Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

Abstract

In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights—which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem, are derived. An explicit and general example is presented to illustrate the theoretical results of the work. The non-Abelian extensions of a family of discrete Painlevé IV equations are discussed.

1. Introduction

Mark Grigorievich Krein [1,2] was the first to discuss matrix extensions of real orthogonal polynomials. Some relevant papers that appear afterwards on this subject are [3,4] and more recently [5]. The Russian mathematicians Aptekarev and Nikishin [5] made a remarkable finding: for a kind of discrete Sturm–Liouville operators they solved the scattering problem and proved that the matrix polynomials that satisfy a three-term recurrence relation with matrix coefficients

$$\begin{array}{cccccc}\hfill x{P}_k\left(x\right)& =A_k{P}_{k+1}\left(x\right)+B_k{P}_k\left(x\right)+A_{k-1}^{*}{P}_{k-1}\left(x\right),\hfill & & & \hfill k& =0,1,\dots ,\hfill \end{array}$$

are orthogonal with respect to a positive definite matrix measure, i.e., they derived a matrix Favard theorem. Later, it was found that matrix orthogonal polynomials (MOP) sometimes satisfy properties, as do the classical orthogonal polynomials.

For example, for matrix versions of Laguerre, Hermite and Jacobi polynomials, the scalar-type Rodrigues’ formula [6,7] and a second order differential equation [8,9,10] has been discussed. It also has been proven in [11] that operators of the form $D={\partial }^2{F}_2\left(t\right)+{\partial }^1{F}_1\left(t\right)+{\partial }^0{F}_0$ have as eigenfunctions different infinite families of MOP’s. In [12,13] matrix extensions of the generalized polynomials considered in [14,15] were studied. Recently, in [16], the Christoffel transformation to matrix orthogonal polynomials in the real line (MOPRL) has been extended and a new matrix Christoffel formula was obtained. Finally, in [17,18] more general transformations—of Geronimus and Uvarov type—where also considered.

Fokas, Its and Kitaev [19] found, in the context of 2D quantum gravity, that certain Riemann–Hilbert problems were solved in terms of orthogonal polynomials in the real line (OPRL). They found that the solution of a $2\times 2$ Riemann–Hilbert problem can be expressed in terms of orthogonal polynomials in the real line and its Cauchy transforms. Later, Deift and Zhou combined these ideas with a non-linear steepest descent analysis in a series of papers [20,21,22,23] which was the seed for a large activity in the field. To mention just a few relevant results, let us cite the study of strong asymptotic with applications in random matrix theory [20,24], the analysis of determinantal point processes [25,26,27,28], orthogonal Laurent polynomials [29,30] and Painlevé equations [31,32].

Recursion coefficients for orthogonal polynomials and their properties is a subject of current interest. See [33,34] for a review on how the form of the weight and its properties translates to the recursion coefficients. Freud [35] has studied weights in $\mathbb{R}$ of exponential variation $w\left(x\right)={\left|x\right|}^{\rho }{exp(-|x|}^m)$, $\rho >-1$ and $m>0$. When $m=2,4,6$ he constructed relations among them as well as determining its asymptotic behavior. The role of the discrete Painlevé I in this context was discovered later by Magnus [36]. For a weight of the form $w\left(\theta \right)=exp(kcos\theta )$, $k\in \mathbb{R}$, on the unit circle the discrete Painlevé II equation for the recursion relations of the corresponding orthogonal polynomials was found in [37,38] (see also [39], for a connection with the Painlevé III equation). The discrete Painlevé II was found in [40] using the Riemann–Hilbert problem given in [41], see also [42]. For a good account of the relation of these discrete Painlevé equations and integrable systems, see [43], and for a survey on the subject of differential and discrete Painlevé equations see (cf. [44]). We also mention the recent paper [45] where a discussion on the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlevé equation can be found. Moreover, in [46], the solution of the discrete alternate Painlevé equations is presented in terms of the Airy function.

The Riemann–Hilbert problem for this matrix situation appears in the paper [47] in the context of inverse scattering for the integral matrix equation, as well as in [48] where the appearance of the Riemann–Hilbert problem for this matrix situation takes place when studying non-Abelian discrete versions of Painlevé I, showing singularity confinement [49], see also [50]. The singularity analysis for a matrix discrete version of the Painlevé I equation was performed. It was found that the singularity confinement holds generically, i.e., in the whole space of parameters except possibly for algebraic subvarieties. This situation was considered in [51] for the matrix extension of the Szegő polynomials in the unit circle and corresponding non-Abelian versions of discrete Painlevé II equations.

In [52], we discussed matrix biorthogonal polynomials with matrix of weights $W\left(z\right)$ such that

• The support of $W\left(z\right)$ is a non-intersecting smooth curve on the complex plane with end points at ∞.
• Weight matrix entries were, in principle, Hölder continuous, and eventually requested to have holomorphic extensions to the complex plane.
• The matrix of weights $W\left(z\right)$ is regular, i.e., $det{\left[W_{j+k}\right]}_{j,k=0,\dots n}\ne 0$, $n\in \mathbb{N}:=\{0,1,\dots \}$, where the moment of order n, $W_n$, associated with W is, for each $n\in \mathbb{N}$, given by, ${\displaystyle W_n:=\frac{1}{2\pi i}{\int }_{\gamma }{z}^{n}W\left(z\right)dz}$.

We obtained Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions, directly from a Riemann–Hilbert problem, with jumps supported on appropriate curves on the complex plane. We considered a Sylvester type differential Pearson equation for the matrix of weights. We also studied whenever the orthogonal polynomials and their second kind functions are solutions of second order linear differential operators with matrix eigenvalues. This was achieved by stating an appropriate boundary value problem for the matrix of weights. In particular, special attention was paid to non-Abelian Hermite biorthogonal polynomials in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with given matrices of degree one polynomial coefficients. We also found nonlinear equations for the matrix coefficients of the corresponding three-term relations, which extends to the non-commutative case the discrete Painlevé I and the alternate discrete Painlevé I equations. In this paper, we conduct a similar study but with more relaxed conditions, namely of Laguerre type.

The layout of the paper is as follows. In Section 2 we present the main definitions and theorems fundamental in the theory worked on in this paper. In Section 3 we state the Riemann–Hilbert problem for matrix biorthogonal polynomials, discussing the Pearson–Laguerre matrix weights with a finite end point, introducing the constant jump fundamental matrix and the important structure matrix. We also apply these ideas to obtain first and second order matrix differential operators of Laguerre type. In Section 4, we take a Laguerre type weight as a case study and reinterpret the results just stated for this specific and general example. Then, in Section 5 we end the paper with the finding of a matrix extension of an instance of the discrete Painlevé IV equation.

2. Preliminaries

2.1. Matrix Biorthogonal Polynomials

We begin this section with the important definition of this paper.

Definition 1

(Laguerre type Matrix of weights). We say that a regular matrix of weights

$$\begin{array}{c}\hfill W=\left[\begin{array}{ccc}{W}^{(1,1)}& \cdots & W^{(1,N)}\\ ⋮& \ddots & ⋮\\ W^{(N,1)}& \cdots & W^{(N,N)}\end{array}\right]\in {\mathbb{C}}^{N\times N}\end{array}$$

is of Laguerre type if:

• The support of $W\left(z\right)$ is a non self-intersecting smooth curve on the complex plane with a beginning point at 0 and an ending point at ∞, and such that it intersects the circles $\left|z\right|=R$, $R\in (0,+\infty )$, once and only once (i.e., it can be taken as a determination curve for $argz$).
• The entries $W^{(j,k)}$ of the matrix measure W can be written as

  $$\begin{array}{cccc}\hfill W^{(j,k)}\left(z\right)& =\sum _{m\in I_{j,k}}{A}_m\left(z\right)z^{{\alpha }_m}{log}^{p_m}z,\hfill & \hfill z& \in \gamma ,\hfill \end{array}$$

  (1)

  where $I_{j,k}$ denotes a finite set of indexes, $Re\left({\alpha }_m\right)>-1$, $p_m\in \mathbb{N}\cup \left\{0\right\}$ and $A_m\left(z\right)$ is Hölder continuous and bounded. Here, the determination of logarithm and the powers are taken along γ. We will request, in the development of the theory, that the functions $A_m$ have a holomorphic extension to the whole complex plane.
• The matrix of weights $W\left(z\right)$ is regular, i.e., $det{\left[W_{j+k}\right]}_{j,k=0,\dots n}\ne 0$, $n\in \mathbb{N}:=\{0,1,\dots \}$, where the moment of order n, $W_n$, associated with W is, for each $n\in \mathbb{N}$, given by, ${\displaystyle W_n:=\frac{1}{2\pi i}{\int }_{\gamma }{z}^{n}W\left(z\right)dz}$.

In this work, for the sake of simplicity, $\gamma =(0,+\infty )$ and the finite end point of the curve $\gamma$ is taken at the origin, $c=0$, with no loss of generality, as similar arguments apply for $c\ne 0$. In [10], different examples of Laguerre weights for the matrix orthogonal polynomials on the real line are studied.

Given a Laguerre type matrix of weights, as in Definition 1, we define the sequences of matrix monic polynomials, ${\left\{P_n^{\mathrm{L}}\left(z\right)\right\}}_{n\in {\mathbb{Z}}_{+}}$, where $deg{P}_n^{\mathrm{L}}\left(z\right)=n$, $n\in \mathbb{N}$, left orthogonal and right orthogonal, ${\left\{P_n^{\mathrm{R}}\left(z\right)\right\}}_{n\in \mathbb{N}}$, where $deg{P}_n^{\mathrm{R}}\left(z\right)=n$, $n\in \mathbb{N}$, with respect to the regular matrix of measure W, by the conditions,

$$\begin{array}{c}\hfill \frac{1}{2\pi i}{\int }_{\gamma }{P}_n^{\mathsf{L}}\left(z\right)W\left(z\right)z^{k}dz={\delta }_{n,k}{C}_n^{-1},\end{array}$$

(2)

$$\begin{array}{c}\hfill \frac{1}{2\pi i}{\int }_{\gamma }{z}^{k}W\left(z\right)P_n^{\mathsf{R}}\left(z\right)dz={\delta }_{n,k}{C}_n^{-1},\end{array}$$

(3)

for $k=0,1,\dots ,n$ and $n\in \mathbb{N}$, where $C_n$ is a nonsingular matrix.

We can see that the sequence of monic polynomials ${\left\{P_n^{\mathrm{L}}\right\}}_{n\in \mathbb{N}}$ are defined by (2) with respect to a regular matrix weight, W. In fact, taking into account a representation for $P_n^{\mathrm{L}}$ as

$$\begin{array}{c}\hfill P_n^{\mathrm{L}}\left(z\right)=p_{\mathrm{L},n}^0{z}^n+p_{\mathrm{L},n}^1{z}^{n-1}+\cdots +p_{\mathrm{L},n}^{n-1}z+p_{\mathrm{L},n}^n,\end{array}$$

such that, for each $j=0,1,\dots ,n-1$

$$\begin{array}{c}\hfill {\int }_{\gamma }{P}_n^{\mathrm{L}}\left(z\right)W\left(z\right)z^j\mathrm{d}z=p_{\mathrm{L},n}^0{W}_{n+j}+p_{\mathrm{L},n}^1{W}_{n+j-1}+\cdots +p_{\mathrm{L},n}^{n-1}{W}_{j+1}+p_{\mathrm{L},n}^n{W}_j=0,\end{array}$$

and with $j=n$

$$\begin{array}{c}\hfill {\int }_{\gamma }{P}_n^{\mathrm{L}}\left(z\right)W\left(z\right)z^n\mathrm{d}z=p_{\mathrm{L},n}^0{W}_{2n}+p_{\mathrm{L},n}^1{W}_{2n-1}+\cdots +p_{\mathrm{L},n}^{n-1}{W}_{n+1}+p_{\mathrm{L},n}^n{W}_n=C_n^{-1}.\end{array}$$

Let us notice that

$$\begin{array}{ccccccc}\hfill {\mathbf{U}}_n={\left[W_{j+k}\right]}_{j,k=0,\dots ,n}=\left[\begin{array}{ccc}{W}_0& \cdots & W_n\\ ⋮& \ddots & ⋮\\ W_n& \cdots & W_{2n}\end{array}\right]& \hfill & \hfill \mathrm{is}\mathrm{such}\mathrm{that}& \hfill & \hfill det{\mathbf{U}}_n\ne 0,& \hfill & \hfill n\in \mathbb{N}.\end{array}$$

In matrix notation, we have

$$\begin{array}{c}\hfill \left[\begin{array}{ccccc}{p}_{\mathrm{L},n}^n& p_{\mathrm{L},n}^{n-1}& \cdots & p_{\mathrm{L},n}^1& p_{\mathrm{L},n}^0\end{array}\right]{\mathbf{U}}_n=\left[\begin{array}{ccccc}0& 0& \cdots & 0& C_n^{-1}\end{array}\right].\end{array}$$

Since $det{\mathbf{U}}_n\ne 0$, we know that the above linear system has a unique solution, i.e., there exists and are unique the matrices $p_{\mathrm{L},n}^n,p_{\mathrm{L},n}^{n-1},\dots ,p_{\mathrm{L},n}^1,p_{\mathrm{L},n}^0$, and so the sequence ${\left\{P_n^{\mathrm{L}}\right\}}_{n\in {\mathbb{Z}}_{+}}$ is uniquely defined up to a multiplicative nonsingular matrix defined by (2).

As a direct consequence of the non-singularity of the last block of $U_n^{-1}$, i.e., the one in the position $(n+1),(n+1)$, of the matrix ${\mathbf{U}}_n^{-1}$ we find that $p_{\mathrm{L},n}^0$ is a non singular matrix. In fact, as (see for instance [53])

$$\begin{array}{c}\hfill {\mathbf{U}}_n^{-1}=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\end{array}$$

with

$$\begin{array}{c}\hfill D={\left(W_{2n}-\left[\begin{array}{ccc}{W}_n& \cdots & W_{2n-1}\end{array}\right]{\mathbf{U}}_{n-1}^{-1}{\left[\begin{array}{ccc}{W}_n^{\top }& \cdots & W_{2n-1}^{\top }\end{array}\right]}^{\top }\right)}^{-1},\end{array}$$

and $detD=\frac{det{\mathbf{U}}_{n-1}}{det{\mathbf{U}}_n}$, we get the non singularity of $p_{\mathrm{L},n}^0$. The same can be seen for ${\left\{P_n^{\mathrm{R}}\right\}}_{n\in \mathbb{N}}$.

The matrix of weights $W\left(z\right)$ induces a non-degenerate bilinear form in the set of matrix polynomials ${\mathbb{C}}^{N\times N}\left[z\right]$ given by

$$\begin{array}{c}\hfill {\langle P,Q\rangle }_W:=\frac{1}{2\pi i}{\int }_{\gamma }P\left(z\right)W\left(z\right)Q\left(z\right)dz,\end{array}$$

(4)

for which ${\left\{P_n^{\mathsf{L}}\left(z\right)\right\}}_{n\in \mathbb{N}}$ and ${\left\{P_n^{\mathsf{R}}\left(z\right)\right\}}_{n\in \mathbb{N}}$ are biorthogonal

$$\begin{array}{cccc}\hfill {〈P_n^{\mathsf{L}},P_m^{\mathsf{R}}〉}_W& ={\delta }_{n,m}{C}_n^{-1},\hfill & \hfill n,m& \in \mathbb{N}.\hfill \end{array}$$

As the polynomials are chosen to be monic, we can write

$$\begin{array}{cc}\hfill P_n^{\mathsf{L}}\left(z\right)& =I_N{z}^n+p_{\mathsf{L},n}^1{z}^{n-1}+p_{\mathsf{L},n}^2{z}^{n-2}+\cdots +p_{\mathsf{L},n}^n,\hfill \\ \hfill P_n^{\mathsf{R}}\left(z\right)& =I_N{z}^n+p_{\mathsf{R},n}^1{z}^{n-1}+p_{\mathsf{R},n}^2{z}^{n-2}+\cdots +p_{\mathsf{R},n}^n,\hfill \end{array}$$

with matrix coefficients $p_{\mathsf{L},n}^k,p_{\mathsf{R},n}^k\in {\mathbb{C}}^{N\times N}$, $k=0,\dots ,n$ and $n\in \mathbb{N}$ (imposing that $p_{\mathsf{L},n}^0=p_{\mathsf{R},n}^0=I_N$, $n\in \mathbb{N}$). Here, $I_N\in {\mathbb{C}}^{N\times N}$ denotes the identity matrix.

We define the sequence of second kind matrix functions by

$$\begin{array}{ccc}\hfill Q_n^{\mathsf{L}}\left(z\right):=\frac{1}{2\pi i}{\int }_{\gamma }\frac{P_n^{\mathsf{L}}\left(z^{\prime }\right)}{z^{\prime }-z}W\left(z^{\prime }\right)d{z}^{\prime },& \hfill & \hfill Q_n^{\mathsf{R}}\left(z\right):=\frac{1}{2\pi i}{\int }_{\gamma }W\left(z^{\prime }\right)\frac{P_n^{\mathsf{R}}\left(z^{\prime }\right)}{z^{\prime }-z}d{z}^{\prime },\end{array}$$

for $n\in \mathbb{N}$. From the orthogonality conditions (2) and (3), we have, for all $n\in \mathbb{N}$, the following asymptotic expansion near infinity

$$\begin{array}{cccc}\hfill Q_n^{\mathsf{L}}\left(z\right)& \sim -C_n^{-1}\left(I_N{z}^{-n-1}+q_{\mathsf{L},n}^1{z}^{-n-2}+\cdots \right),\hfill & \hfill \left|z\right|& \to \infty ,\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill Q_n^{\mathsf{R}}\left(z\right)& \sim -\left(I_N{z}^{-n-1}+q_{\mathsf{R},n}^1{z}^{-n-2}+\cdots \right)C_n^{-1},\hfill & \hfill \left|z\right|& \to \infty .\hfill \end{array}$$

(6)

2.2. Three-Term Recurrence Relation

Following standard arguments of orthogonality, we conclude that the sequence of monic polynomials ${\left\{P_n^{\mathsf{L}}\left(z\right)\right\}}_{n\in \mathbb{N}}$ satisfies the three-term recurrence relations

$$\begin{array}{cccc}\hfill z{P}_n^{\mathsf{L}}\left(z\right)& =P_{n+1}^{\mathsf{L}}\left(z\right)+{\beta }_n^{\mathsf{L}}{P}_n^{\mathsf{L}}\left(z\right)+{\gamma }_n^{\mathsf{L}}{P}_{n-1}^{\mathsf{L}}\left(z\right),\hfill & \hfill & n\in \mathbb{N},\hfill \\ \hfill z{Q}_n^{\mathrm{L}}\left(z\right)& =Q_{n+1}^{\mathrm{L}}\left(z\right)+{\beta }_n^{\mathrm{L}}{Q}_n^{\mathrm{L}}\left(z\right)+{\gamma }_n^{\mathsf{L}}{Q}_{n-1}^{\mathrm{L}}\left(z\right),\hfill & \hfill & n\in \mathbb{N},\hfill \end{array}$$

with recursion coefficients given by ${\beta }_n^{\mathsf{L}}:=p_{\mathsf{L},n}^1-p_{\mathsf{L},n+1}^1$ and ${\gamma }_n^{\mathsf{L}}:=C_n^{-1}{C}_{n-1}$, with initial conditions,

$$\begin{array}{ccccccccc}\hfill P_{-1}^{\mathsf{L}}=0_N,& \hfill & \hfill P_0^{\mathsf{L}}=I_N,& \hfill & \hfill Q_{-1}^{\mathrm{L}}\left(z\right)=-C_{-1}^{-1}& \hfill & \hfill \mathrm{and}& \hfill & \hfill Q_0^{\mathrm{L}}\left(z\right)=S_W\left(z\right):=\frac{1}{2\pi \mathrm{i}}{\int }_{\gamma }\frac{W\left(z^{\prime }\right)}{z^{\prime }-z}\mathrm{d}{z}^{\prime },\end{array}$$

where $S_W\left(z\right)$ is the Stieltjes–Markov transformation. Analogously,

$$\begin{array}{cccc}\hfill z{P}_n^{\mathsf{R}}\left(z\right)& =P_{n+1}^{\mathsf{R}}\left(z\right)+P_n^{\mathsf{R}}\left(z\right){\beta }_n^{\mathsf{R}}+P_{n-1}^{\mathsf{R}}\left(z\right){\gamma }_n^{\mathsf{R}},\hfill & \hfill & n\in \mathbb{N},\hfill \\ \hfill z{Q}_n^{\mathsf{R}}\left(z\right)& =Q_{n+1}^{\mathsf{R}}\left(z\right)+Q_n^{\mathsf{R}}\left(z\right){\beta }_n^{\mathsf{R}}+Q_{n-1}^{\mathsf{R}}\left(z\right){\gamma }_n^{\mathsf{R}},\hfill & \hfill & n\in \mathbb{N},\hfill \end{array}$$

where ${\beta }_n^{\mathsf{R}}:=C_n{\beta }_n^{\mathsf{L}}{C}_n^{-1}$, ${\gamma }_n^{\mathsf{R}}:=C_n{\gamma }_n^{\mathsf{L}}{C}_n^{-1}=C_{n-1}{C}_n^{-1}$, $Q_{-1}^{\mathrm{R}}\left(z\right)=-C_{-1}^{-1}$ and $Q_0^{\mathrm{R}}\left(z\right)=S_W\left(z\right)$.

These relations could be written as,

$$\begin{array}{cccccc}\hfill Y_{n+1}^{\mathsf{L}}\left(z\right)& =T_n^{\mathsf{L}}\left(z\right)Y_n^{\mathsf{L}}\left(z\right),\hfill & \hfill T_n^{\mathsf{L}}\left(z\right)& :=\left[\begin{array}{cc}z{I}_N-{\beta }_n^{\mathsf{L}}& C_n^{-1}\\ -C_n& 0_N\end{array}\right],\hfill & \hfill n& \in \mathbb{N},\hfill \end{array}$$

where $T_n^{\mathsf{L}}$ denotes the left transfer matrix. For the right orthogonality, we similarly obtain,

$$\begin{array}{cccccc}\hfill Y_{n+1}^{\mathsf{R}}\left(z\right)& =Y_n^{\mathsf{R}}\left(z\right)T_n^{\mathsf{R}}\left(z\right),\hfill & \hfill T_n^{\mathsf{R}}\left(z\right)& :=\left[\begin{array}{cc}z{I}_N-{\beta }_n^{\mathsf{R}}& -C_n\\ C_n^{-1}& 0_N\end{array}\right],\hfill & \hfill n& \in \mathbb{N},\hfill \end{array}$$

where $T_n^{\mathsf{L}}$ denotes the right transfer matrix.

2.3. Reductions: From Biorthogonality to Orthogonality

We consider two possible reductions for the matrix of weights, the symmetric reduction and the Hermitian reduction. (i) A matrix of weights $W\left(z\right)$ with support on $\gamma$ is said to be symmetric if

$$\begin{array}{ccc}\hfill {\left(W\left(z\right)\right)}^{\top }=W\left(z\right),& \hfill & \hfill z\in \gamma .\end{array}$$

(ii) A matrix of weights $W\left(x\right)$ with support on $\mathbb{R}$ is said to be Hermitian if

$$\begin{array}{ccc}\hfill {\left(W\left(x\right)\right)}^{†}=W\left(x\right),& \hfill & \hfill x\in \mathbb{R}.\end{array}$$

These two reductions lead to orthogonal polynomials, as the two biorthogonal families are identified; i.e., for the symmetric case

$$\begin{array}{ccccc}\hfill P_n^R\left(z\right)={\left(P_n^{\mathrm{L}}\left(z\right)\right)}^{\top },& \hfill & \hfill Q_n^{\mathrm{R}}\left(z\right)={\left(Q_n^{\mathrm{L}}\left(z\right)\right)}^{\top },& \hfill & \hfill z\in \mathbb{C},\end{array}$$

and for the Hermitian case, with $\gamma =\mathbb{R}$

$$\begin{array}{ccccc}\hfill P_n^{\mathrm{R}}\left(z\right)={\left(P_n^{\mathrm{L}}\left(\overline{z}\right)\right)}^{†},& \hfill & \hfill Q_n^{\mathrm{R}}\left(z\right)={\left(Q_n^{\mathrm{L}}\left(\overline{z}\right)\right)}^{†},& \hfill & \hfill z\in \mathbb{C}.\end{array}$$

In both cases, biorthogonality collapses into orthogonality, that for the symmetric case reads as

$$\begin{array}{ccc}\hfill \frac{1}{2\pi \mathrm{i}}{\int }_{\gamma }{P}_n\left(z\right)W\left(z\right){\left(P_m\left(z\right)\right)}^{\top }\mathrm{d}z={\delta }_{n,m}{C}_n^{-1},& \hfill & \hfill n,m\in \mathbb{N},\end{array}$$

while for the Hermitian case it can be written as follows

$$\begin{array}{ccc}\hfill \frac{1}{2\pi \mathrm{i}}{\int }_{\mathbb{R}}{P}_n\left(x\right)W\left(x\right){\left(P_m\left(x\right)\right)}^{†}\mathrm{d}x={\delta }_{n,m}{C}_n^{-1},& \hfill & \hfill n,m\in \mathbb{N},\end{array}$$

where $P_n=P_n^{\mathrm{L}}$.

3. Riemann–Hilbert Problem for Matrix Biorthogonal Polynomials

3.1. The Riemann–Hilbert Problem

We begin this section stating a general theorem on the Riemann–Hilbert problem for the matrix Laguerre general weights. A preliminary version of this can be found in [54].

Theorem 1.

Given a regular Laguerre type matrix of weights $W\left(x\right)$ with support on γ, we have:

(i) 

The matrix function

$$\begin{array}{cc}\hfill Y_n^{\mathsf{L}}\left(z\right)& :=\left[\begin{array}{cc}{P}_n^{\mathsf{L}}\left(z\right)& Q_n^{\mathsf{L}}\left(z\right)\\ -C_{n-1}{P}_{n-1}^{\mathsf{L}}\left(z\right)& -C_{n-1}{Q}_{n-1}^{\mathsf{L}}\left(z\right)\end{array}\right]\hfill \end{array}$$

is, for each $n\in \mathbb{N}$, the unique solution of the Riemann–Hilbert problem, which consists in the determination of a $2N\times 2N$ complex matrix function such that:

(RHL1)

$Y_n^{\mathsf{L}}\left(z\right)$ is holomorphic in $\mathbb{C}\setminus \gamma$.

(RHL2)

Has the following asymptotic behavior near infinity,

$$\begin{array}{c}\hfill Y_n^{\mathsf{L}}\left(z\right)\sim \left(I_{2N}+\sum _{j=1}^{\infty }\left(z^{-j}\right)Y_n^{j,\mathsf{L}}\right)\left[\begin{array}{cc}{I}_N{z}^n& 0_N\\ 0_N& I_N{z}^{-n}\end{array}\right].\end{array}$$

(RHL3)

Satisfies the jump condition

$$\begin{array}{cccc}\hfill {\left(Y_n^{\mathsf{L}}\left(z\right)\right)}_{+}& ={\left(Y_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}{I}_N& W\left(z\right)\\ 0_N& I_N\end{array}\right],\hfill & \hfill z& \in \gamma \setminus \left\{0\right\}.\hfill \end{array}$$

(RHL4)

$Y_n^{\mathsf{L}}\left(z\right)=\left[\begin{array}{cc}O\left(1\right)& s_1^{\mathsf{L}}\left(z\right)\\ O\left(1\right)& s_2^{\mathsf{L}}\left(z\right)\end{array}\right]$, as $z\to 0$, ${\displaystyle \underset{z\to 0}{lim}z{s}_j^{\mathsf{L}}\left(z\right)=0_N}$, $j=1,2$ and the O conditions are understood entrywise.

(ii) 

The matrix function

$$\begin{array}{cc}\hfill Y_n^{\mathsf{R}}\left(z\right)& :=\left[\begin{array}{cc}{P}_n^{\mathsf{R}}\left(z\right)& -P_{n-1}^{\mathsf{R}}\left(z\right)C_{n-1}\\ Q_n^{\mathsf{R}}\left(z\right)& -Q_{n-1}^{\mathsf{R}}\left(z\right)C_{n-1}\end{array}\right]\hfill \end{array}$$

is, for each $n\in \mathbb{N}$, the unique solution of the Riemann–Hilbert problem, which consists of the determination of a $2N\times 2N$ complex matrix function such that:

(RHR1)

$Y_n^{\mathsf{R}}\left(z\right)$ is holomorphic in $\mathbb{C}\setminus \gamma$.

(RHR2)

Has the following asymptotic behavior near infinity,

$$\begin{array}{c}\hfill Y_n^{\mathsf{R}}\left(z\right)\sim \left[\begin{array}{cc}{I}_N{z}^n& 0_N\\ 0_N& I_N{z}^{-n}\end{array}\right]\left(I_{2N}+\sum _{j=1}^{\infty }\left(z^{-j}\right)Y_n^{j,\mathsf{R}}\right).\end{array}$$

(RHR3)

Satisfies the jump condition

$$\begin{array}{cccc}\hfill {\left(Y_n^{\mathsf{R}}\left(z\right)\right)}_{+}& =\left[\begin{array}{cc}{I}_N& 0_N\\ W\left(z\right)& I_N\end{array}\right]{\left(Y_n^{\mathsf{R}}\left(z\right)\right)}_{-},\hfill & \hfill z& \in \gamma \setminus \left\{0\right\}.\hfill \end{array}$$

(RHR4)

$Y_n^{\mathsf{R}}\left(z\right)=\left[\begin{array}{cc}O\left(1\right)& O\left(1\right)\\ s_1^{\mathsf{R}}\left(z\right)& s_2^{\mathsf{R}}\left(z\right)\end{array}\right]$, as $z\to 0$, ${\displaystyle \underset{z\to 0}{lim}z{s}_j^{\mathsf{R}}\left(z\right)=0_N}$, $j=1,2$ and the O conditions are understood entrywise.

(iii) 

The determinant of $Y_n^{\mathsf{L}}\left(z\right)$ and $Y_n^{\mathsf{R}}\left(z\right)$ are both equal to 1, for every $z\in \mathbb{C}$.

Proof. 

Using the standard calculations from the scalar case [55], it follows that the matrices $Y_n^{\mathsf{L}}$ and $Y_n^{\mathsf{R}}$ satisfy $\left(\mathrm{RHL}1\right)$–$\left(\mathrm{RHL}3\right)$ and $\left(\mathrm{RHR}1\right)$–$\left(\mathrm{RHR}3\right),$ respectively.

The entries $W^{j,k}$ of the matrix measure W are given in (1). It holds (cf. [56]) that in a neighborhood of $z=0$ the Cauchy transform

$$\begin{array}{c}\hfill {\varphi }_m\left(z\right)=\frac{1}{2\pi i}{\int }_{\gamma }\frac{p\left(\zeta \right)A_m\left(\zeta \right){\zeta }^{{\alpha }_m}{log}^{p_m}\zeta }{\zeta -z}d\zeta ,\end{array}$$

where $p\left(\zeta \right)$ denotes any polynomial in $\zeta$, that satisfies ${\displaystyle \underset{z\to 0}{lim}z{\varphi }_m\left(z\right)=0}$. Then, $\left(\mathrm{RHL}4\right)$ and $\left(\mathrm{RHR}4\right)$ are fulfilled by the matrices $Y_n^{\mathsf{L}},Y_n^{\mathsf{R}}$, respectively. To prove the unicity of both Riemann–Hilbert problems let us consider the matrix function

$$\begin{array}{c}\hfill G\left(z\right)=Y_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{0}_N& I_N\\ -I_N& 0_N\end{array}\right]Y_n^{\mathsf{R}}\left(z\right)\left[\begin{array}{cc}{0}_N& -I_N\\ I_N& 0_N\end{array}\right].\end{array}$$

It can easily be proved that $G\left(z\right)$ has no jump or discontinuity on the curve $\gamma$ and that its behavior at the end point 0 is given by

$$\begin{array}{cccc}\hfill G\left(z\right)& \sim \left[\begin{array}{cc}O\left(1\right)s_1^{\mathsf{L}}\left(z\right)+O\left(1\right)s_2^{\mathsf{R}}\left(z\right)& O\left(1\right)s_1^{\mathsf{L}}\left(z\right)+O\left(1\right)s_1^{\mathsf{R}}\left(z\right)\\ O\left(1\right)s_2^{\mathsf{L}}\left(z\right)+O\left(1\right)s_2^{\mathsf{R}}\left(z\right)& O\left(1\right)s_2^{\mathsf{L}}\left(z\right)+O\left(1\right)s_1^{\mathsf{R}}\left(z\right)\end{array}\right],\hfill & \hfill z& \to 0,\hfill \end{array}$$

so it holds that ${\displaystyle \underset{z\to 0}{lim}zG\left(z\right)=0}$ and we conclude that the end point 0 is a removable singularity of G. Now, from the behavior for $z\to \infty$,

$$\begin{array}{c}\hfill G\left(z\right)\sim \left[\begin{array}{cc}{I}_N{z}^n& 0_N\\ 0_N& I_N{z}^{-n}\end{array}\right]\left[\begin{array}{cc}{0}_N& I_N\\ -I_N& 0_N\end{array}\right]\left[\begin{array}{cc}{I}_N{z}^n& 0_N\\ 0_N& I_N{z}^{-n}\end{array}\right]\left[\begin{array}{cc}{0}_N& -I_N\\ I_N& 0_N\end{array}\right]=\left[\begin{array}{cc}{I}_N& 0_N\\ 0_N& I_N\end{array}\right],\end{array}$$

hence the Liouville theorem implies that $G\left(z\right)=I_{2N}$. To prove the unicity of the solution of $\left(\mathrm{RHL}1\right)$–$\left(\mathrm{RHL}3\right)$ and $\left(\mathrm{RHR}1\right)$–$\left(\mathrm{RHR}3\right)$ let ${{\tilde{Y}}^{\mathsf{L}}}_n$ be any solution of the left Riemann–Hilbert problem. Then

$$\begin{array}{c}\hfill {{\tilde{Y}}^{\mathsf{L}}}_n\left(z\right)={\left(\left[\begin{array}{cc}{0}_N& I_N\\ -I_N& 0_N\end{array}\right]Y_n^{\mathsf{R}}\left(z\right)\left[\begin{array}{cc}{0}_N& -I_N\\ I_N& 0_N\end{array}\right]\right)}^{-1}.\end{array}$$

Hence, any solution of this left Riemann–Hilbert problem is equal to the inverse of a fixed matrix, and the uniqueness follows. We obtain the uniqueness of the solution of the right Riemann–Hilbert in a similar way.

Let us calculate the determinant of the fundamental matrix. Since

$$\begin{array}{ccc}\hfill det\left(Y_{n+1}^{\mathsf{L}}\right)=det\left(\left[\begin{array}{cc}z{I}_N-{\beta }_n^{\mathsf{L}}& C_n^{-1}\\ -C_n& 0_N\end{array}\right]\right)det\left(Y_n^{\mathsf{L}}\left(z\right)\right)=det\left(Y_n^{\mathsf{L}}\right),& \hfill & \hfill \forall n\in \mathbb{N}.\end{array}$$

Then

$$\begin{array}{c}\hfill det\left(Y_n^{\mathsf{L}}\right)=det\left(Y_0^{\mathsf{L}}\right)=det\left[\begin{array}{cc}{P}_0^{\mathsf{L}}\left(z\right)& Q_0^{\mathsf{L}}\left(z\right)\\ -C_{-1}{P}_{-1}^{\mathsf{L}}\left(z\right)& -C_{-1}{Q}_{-1}^{\mathsf{L}}\left(z\right)\end{array}\right]=1.\end{array}$$

A similar reasoning leads to a similar result for $Y_n^{\mathsf{R}}$. □

Corollary 1.

It holds that

$$\begin{array}{c}\hfill {\left(Y_n^{\mathsf{L}}\left(z\right)\right)}^{-1}=\left[\begin{array}{cc}{0}_N& I_N\\ -I_N& 0_N\end{array}\right]Y_n^{\mathsf{R}}\left(z\right)\left[\begin{array}{cc}{0}_N& -I_N\\ I_N& 0_N\end{array}\right],\end{array}$$

that entrywise reads as follows

$$\begin{array}{cc}\hfill Q_n^{\mathsf{L}}\left(z\right)P_{n-1}^{\mathsf{R}}\left(z\right)-P_n^{\mathsf{L}}\left(z\right)Q_{n-1}^{\mathsf{R}}\left(z\right)& =C_{n-1}^{-1},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill P_{n-1}^{\mathsf{L}}\left(z\right)Q_n^{\mathsf{R}}\left(z\right)-Q_{n-1}^{\mathsf{L}}\left(z\right)P_n^{\mathsf{R}}\left(z\right)& =C_{n-1}^{-1},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill Q_n^{\mathsf{L}}\left(z\right)P_n^{\mathsf{R}}\left(z\right)-P_n^{\mathsf{L}}\left(z\right)Q_n^{\mathsf{R}}\left(z\right)& =0.\hfill \end{array}$$

(9)

3.2. Pearson–Laguerre Matrix Weights with a Finite End Point

Instead of a given matrix of weights, we consider two matrices of entire functions, say $h^{\mathsf{L}}\left(z\right)$ and $h^{\mathsf{R}}\left(z\right)$, such that the following matrix Pearson equations are satisfied

$$\begin{array}{ccc}\hfill z{\left(W^{\mathsf{L}}\right)}^{\prime }\left(z\right)=h^{\mathsf{L}}\left(z\right)W^{\mathsf{L}}\left(z\right),& \hfill & \hfill z{\left(W^{\mathsf{R}}\right)}^{\prime }\left(z\right)=W^{\mathsf{R}}\left(z\right)h^{\mathsf{R}}\left(z\right),\end{array}$$

(10)

and, given solutions to them, we construct the corresponding matrix of weights as $W=W^{\mathsf{L}}{W}^{\mathsf{R}}$. This matrix of weights is also characterized by a Pearson equation.

Proposition 1

(Pearson differential equation). Given two matrices of entire functions $h^{\mathsf{L}}\left(z\right)$ and $h^{\mathsf{R}}\left(z\right)$. A solution of the Sylvester type matrix differential equation, which we call a Pearson equation for the weight, W,

$$\begin{array}{c}\hfill z{W}^{\prime }\left(z\right)=h^{\mathsf{L}}\left(z\right)W\left(z\right)+W\left(z\right)h^{\mathsf{R}}\left(z\right),\end{array}$$

(11)

is of the form $W=W^{\mathsf{L}}{W}^{\mathsf{R}}$ where the factor matrices $W^{\mathsf{L}}$ and $W^{\mathsf{R}}$ are solutions of (10).

Proof. 

Given solutions $W^{\mathsf{L}}$ and $W^{\mathsf{R}}$ of (10), it follows immediately, just using the Leibniz law for derivatives, that $W=W^{\mathsf{L}}{W}^{\mathsf{R}}$ fulfills (11). Moreover, given a solution W of (11) we pick a solution $W^{\mathsf{L}}$ of the first equation in (10), then it is easy to see that ${\left(W^{\mathsf{L}}\right)}^{-1}W$ satisfies the second equation in (10). □

We can give the following result from the literature [57].

Theorem 2

(Solution at a regular singular point). Let $h^{\mathsf{L}}\left(z\right)$ be an entire matrix function. Then, for the solutions of the Pearson Equation (10) we have:

(i) 

If $A^{\mathsf{L}}:=h^{\mathsf{L}}\left(0\right)$ has no eigenvalues that differ from each other by positive integers then, the solution of the left matrix differential equation in (10) can be written as

$$\begin{array}{c}\hfill W^{\mathsf{L}}\left(z\right)=H^{\mathsf{L}}\left(z\right)z^{A^{\mathsf{L}}}{W}_0^{\mathsf{L}},\end{array}$$

where $H^{\mathsf{L}}\left(z\right)$ is an entire and nonsingular matrix function such that $H^{\mathsf{L}}\left(0\right)=I_N$, and $W_0^{\mathsf{L}}$ is a constant nonsingular matrix.

(ii)

If the matrix function $A^{\mathsf{L}}$ has eigenvalues that differ from each other by positive integers, then the solution of the left matrix differential equation in (10) can be written as

$$\begin{array}{c}\hfill W^{\mathsf{L}}\left(z\right)=H^{\mathsf{L}}\left(z\right)z^{{\tilde{A}}^{\mathsf{L}}}{W}_0^{\mathsf{L}},\end{array}$$

where, in this case,

$$\begin{array}{c}\hfill H^{\mathsf{L}}\left(z\right)={\tilde{S}}^{\mathsf{L}}\left(z\right){\Pi }^{\mathsf{L}}\left(z\right),\end{array}$$

and ${\tilde{S}}^{\mathsf{L}}\left(z\right)$ is a finite product of factors of the form $T_i{S}_i^{\mathsf{L}}\left(z\right)$, with $T_i$ a nonsingular matrix and $S_i^{\mathsf{L}}\left(z\right)$ is a shearing matrix, i.e., a matrix given by blocks as

$$\begin{array}{c}\hfill S_i^{\mathsf{L}}\left(z\right)=\left[\begin{array}{cc}{I}_{n_i}& 0\\ 0& z{I}_{m_i}\end{array}\right],\end{array}$$

for some positive integers $n_i,m_i$, and ${\Pi }^{\mathsf{L}}\left(z\right)$ is an entire and non singular matrix function such that ${\Pi }^{\mathsf{L}}\left(0\right)=I$, ${\tilde{A}}^{\mathsf{L}}$ is a constant matrix built from the matrix $A^{\mathsf{L}}$, where the eigenvalues of this matrix are decreased in such a way that the eigenvalues of the resulting matrix do not differ by a positive integer and $W_0^{\mathsf{L}}$ is a constant nonsingular matrix.

We can obtain analogous results for the right matrix differential equation in (10) and we will denote the solution as

$$\begin{array}{c}\hfill W^{\mathsf{R}}\left(z\right)=W_0^{\mathsf{R}}{z}^{A^{\mathsf{R}}}{H}^{\mathsf{R}}\left(z\right).\end{array}$$

Notice that given a matrix A, and the oriented curve $\gamma$, the matrix of functions $z^A=e^{Alogz}$ is a matrix of holomorphic functions in $\mathbb{C}\setminus \gamma$, and

$$\begin{array}{cccc}\hfill {\left(z^A\right)}_{-}& ={\left(z^A\right)}_{+}{e}^{2\pi iA}=e^{2\pi iA}{\left(z^A\right)}_{+},\hfill & \hfill z& \in \gamma .\hfill \end{array}$$

We also adopt the convention that ${\left(W^{\mathsf{L}}\left(z\right)W^{\mathsf{R}}\left(z\right)\right)}_{+}=W\left(z\right)$, i.e., the matrix of weight is obtained from the limit behavior of the right side of the curve $\gamma$ of the matrix function $W^{\mathsf{L}}\left(z\right)W^{\mathsf{R}}\left(z\right)$.

It is necessary, in order to consider the Riemann–Hilbert problem related to the matrix of weights W satisfying (11), to study the behavior of $W\left(z\right)$ around the origin. For that aim, let us consider J, the Jordan matrix similar to the matrix A, i.e., there exists an nonsingular matrix P such that $A=PJ{P}^{-1}$. It holds $z^A=P{z}^J{P}^{-1}$ so if

$$\begin{array}{c}\hfill J=({\lambda }_1{I}_{m_1}+N_1)\oplus ({\lambda }_2{I}_{m_2}+N_2)\oplus \cdots \oplus ({\lambda }_s{I}_{m_s}+N_s)\end{array}$$

where $m_k$ is the order of the nilpotent matrix $N_k$, we have that

$$\begin{array}{c}\hfill z^J=z^{{\lambda }_1{I}_{m_1}+N_1}\oplus z^{{\lambda }_2{I}_{m_2}+N_2}\oplus \cdots \oplus z^{{\lambda }_s{I}_{m_s}+N_s}\end{array}$$

where $z^{{\lambda }_k{I}_{m_k}+N_k}=z^{{\lambda }_k{I}_{m_k}}{z}^{N_k}$. It is straightforward that $z^{{\lambda }_k{I}_{m_k}}=z^{{\lambda }_k}{I}_{m_k}$ and

$$\begin{array}{c}\hfill z^{N_k}=e^{N_{k}logz}=I_{m_k}+logz{N}_k+\frac{{log}^{2}z}{2!}{N}_k^2+\cdots +\frac{{log}^{m_k-1}z}{(m_k-1)!}{N}_k^{m_k-1},\end{array}$$

where we have used the nilpotency of $N_k^j=0_N$ for $j\ge m_k$. So we can conclude that the entries of $z^A$ are linear combinations of $z^{{\lambda }_j}$ with polynomials coefficients in the variable $logz$.

3.3. Constant Jump Fundamental Matrix

According to the above notation and given a regular matrix of weights as described in (11), we introduce the constant jump fundamental matrices

$$\begin{array}{cc}\hfill Z_n^{\mathsf{L}}\left(z\right)& :=Y_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{W}^{\mathsf{L}}\left(z\right)& 0_N\\ 0_N& {\left(W^{\mathsf{R}}\left(z\right)\right)}^{-1}\end{array}\right],\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill Z_n^{\mathsf{R}}\left(z\right)& :=\left[\begin{array}{cc}{W}^{\mathsf{R}}\left(z\right)& 0_N\\ 0_N& {\left(W^{\mathsf{L}}\left(z\right)\right)}^{-1}\end{array}\right]Y_n^{\mathsf{R}}\left(z\right),\hfill \end{array}$$

(13)

for $n\in \mathbb{N}$.

Proposition 2.

The constant jump fundamental matrices $Z_n^{\mathsf{L}}\left(z\right)$ and $Z_n^{\mathsf{R}}\left(z\right)$ satisfy, for each $n\in \mathbb{N}$, the following properties:

(i)

They are holomorphic on $\mathbb{C}\setminus \gamma$.

(ii) 

Present the following constant jump condition on γ

$$\begin{array}{cc}\hfill {\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{+}& ={\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}(W_0^{\mathsf{L}}{)}^{-1}{e}^{-2\pi i{A}^{\mathsf{L}}}{W}_0^{\mathsf{L}}& (W_0^{\mathsf{L}}{)}^{-1}{e}^{-2\pi i{A}^{\mathsf{L}}}{W}_0^{\mathsf{L}}\\ 0_N& W_0^{\mathsf{R}}{e}^{2\pi i{A}^{\mathsf{R}}}{\left(W_0^{\mathsf{R}}\right)}^{-1}\end{array}\right],\hfill \\ \hfill {\left(Z_n^{\mathsf{R}}\left(z\right)\right)}_{+}& =\left[\begin{array}{cc}{W}_0^{\mathsf{R}}{e}^{-2\pi i{A}^{\mathsf{R}}}{\left(W_0^{\mathsf{R}}\right)}^{-1}& 0_N\\ W_0^{\mathsf{R}}{e}^{-2\pi i{A}^{\mathsf{R}}}({W_0^{\mathsf{R}})}^{-1}& {W_0^{\mathsf{L}}}^{-1}{e}^{2\pi i{A}^{\mathsf{L}}}{W}_0^{\mathsf{L}}\end{array}\right]{\left(Z_n^{\mathsf{R}}\left(z\right)\right)}_{-},\hfill \end{array}$$

for all $z\in \gamma$.

Proof. 

(i) The holomorphic properties of $Z_n^{\mathsf{L}}$ are inherited from those of the fundamental matrices $Y_n^{\mathsf{L}}$ and $z^A$ and taking into account that $H^{\mathsf{L}}\left(z\right)$ is an entire matrix function.

(ii) From the definition of $Z_n^{\mathsf{L}}\left(z\right)$, we have

$$\begin{array}{c}\hfill {\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{+}={\left(Y_n^{\mathsf{L}}\left(z\right)\right)}_{+}\left[\begin{array}{cc}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{+}& 0_N\\ 0_N& {\left(W^{\mathsf{R}}\left(z\right)\right)}_{+}^{-1}\end{array}\right],\end{array}$$

and taking into account Theorem 1 we successively obtain

$$\begin{array}{cc}\hfill & {\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{+}={\left(Y_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}{I}_N& {\left(W^{\mathsf{L}}\left(z\right)W^{\mathsf{R}}\left(z\right)\right)}_{+}\\ 0_N& I_N\end{array}\right]\left[\begin{array}{cc}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{+}& 0_N\\ 0_N& {\left(W^{\mathsf{R}}\left(z\right)\right)}_{+}^{-1}\end{array}\right]\hfill \\ \hfill & \phantom{olaaa}={\left(Y_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{-}& 0_N\\ 0_N& {\left(W^{\mathsf{R}}\left(z\right)\right)}_{-}^{-1}\end{array}\right]\left[\begin{array}{cc}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{-}^{-1}& 0_N\\ 0_N& {\left(W^{\mathsf{R}}\left(z\right)\right)}_{-}\end{array}\right]\hfill \\ \hfill & \phantom{olaolaolaolaolaolaolaolaolaolaolaolaolaolaolaolaola}\times \left[\begin{array}{cc}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{+}& {\left(W^{\mathsf{L}}\left(z\right)\right)}_{+}\\ 0_N& {\left(W^{\mathsf{R}}\left(z\right)\right)}_{+}^{-1}\end{array}\right]\hfill \\ \hfill & \phantom{olaaa}={\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{-}^{-1}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{+}& {\left(W^{\mathsf{L}}\left(z\right)\right)}_{-}^{-1}{\left(W^{\mathsf{L}}\left(z\right)\right)}_{+}\\ 0_N& {\left(W^{\mathsf{R}}\left(z\right)\right)}_{-}{\left(W^{\mathsf{R}}\left(z\right)\right)}_{+}^{-1}\end{array}\right]\hfill \\ \hfill & \phantom{olaaa}={\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}{\left(W_0^{\mathsf{L}}\right)}^{-1}{e}^{-2\pi i{A}^{\mathsf{L}}}{W}_0^{\mathsf{L}}& {\left(W_0^{\mathsf{L}}\right)}^{-1}{e}^{-2\pi i{A}^{\mathsf{L}}}{W}_0^{\mathsf{L}}\\ 0_N& W_0^{\mathsf{R}}{e}^{2\pi i{A}^{\mathsf{L}}}{\left(W_0^{\mathsf{R}}\right)}^{-1}\end{array}\right].\hfill \end{array}$$

Hence, we obtain the desired constant jump condition for $Z_n^{\mathsf{L}}\left(z\right)$.

To complete the proof we only have to check that

$$\begin{array}{c}\hfill Z_n^{\mathsf{R}}\left(z\right)=\left[\begin{array}{cc}0& -I_N\\ I_N& 0\end{array}\right]{\left(Z_n^{\mathsf{L}}\left(z\right)\right)}^{-1}\left[\begin{array}{cc}0& I_N\\ -I_N& 0\end{array}\right],\end{array}$$

(14)

which is a consequence of (13). □

3.4. Structure Matrix and Zero Curvature Formula

In parallel to the matrices $Z_n^{\mathsf{L}}\left(z\right)$ and $Z_n^{\mathsf{R}}\left(z\right)$, for each factorization we introduce what we call structure matrices given in terms of the left, respectively right, logarithmic derivatives by,

$$\begin{array}{cccc}\hfill M_n^{\mathsf{L}}\left(z\right)& :={\left(Z_n^{\mathsf{L}}\right)}^{\prime }\left(z\right){\left(Z_n^{\mathsf{L}}\left(z\right)\right)}^{-1},\hfill & \hfill M_n^{\mathsf{R}}\left(z\right)& :={\left(Z_n^{\mathsf{R}}\left(z\right)\right)}^{-1}{\left(Z_n^{\mathsf{R}}\right)}^{\prime }\left(z\right).\hfill \end{array}$$

(15)

It is not difficult to prove that

$$\begin{array}{ccc}\hfill M_n^{\mathsf{R}}\left(z\right)& =-\left[\begin{array}{cc}0& -I_N\\ I_N& 0\end{array}\right]M_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}0& I_N\\ -I_N& 0\end{array}\right],\hfill & \hfill n\in \mathbb{N}.\end{array}$$

(16)

Proposition 3

([52]). We have the following properties:

(i) 

The transfer matrices satisfy

$$\begin{array}{ccccc}\hfill T_n^{\mathsf{L}}\left(z\right)Z_n^{\mathsf{L}}\left(z\right)& =Z_{n+1}^{\mathsf{L}}\left(z\right),\hfill & \hfill Z_n^{\mathsf{R}}\left(z\right)T_n^{\mathsf{R}}\left(z\right)& =Z_{n+1}^{\mathsf{R}}\left(z\right),\hfill & \hfill n\in \mathbb{N}.\end{array}$$

(ii) 

The zero curvature formulas

$$\begin{array}{cc}\hfill \left[\begin{array}{cc}{I}_N& 0_N\\ 0_N& 0_N\end{array}\right]& =M_{n+1}^{\mathsf{L}}\left(z\right)T_n^{\mathsf{L}}\left(z\right)-T_n^{\mathsf{L}}\left(z\right)M_n^{\mathsf{L}}\left(z\right),\hfill \\ \hfill \left[\begin{array}{cc}{I}_N& 0_N\\ 0_N& 0_N\end{array}\right]& =T_n^{\mathsf{R}}\left(z\right)M_{n+1}^{\mathsf{R}}\left(z\right)-M_n^{\mathsf{R}}\left(z\right)T_n^{\mathsf{R}}\left(z\right),\hfill \end{array}$$

$n\in \mathbb{N}$, are fulfilled.

Now, we discuss the holomorphic properties of the structure matrices just introduced.

Theorem 3.

The structure matrices $M_n^{\mathsf{L}}\left(z\right)$ and $M_n^{\mathsf{R}}\left(z\right)$, cf. (15), are, for each $n\in \mathbb{N}$, meromorphic on $\mathbb{C}$, with singularity located at $z=0$, which happens to be a removable singularity or a simple pole.

Proof. 

Let us prove the statement for $M_n^{\mathsf{L}}\left(z\right)$, for $M_n^{\mathsf{R}}\left(z\right)$ one should proceed similarly. From (15) it follows that $M_n^{\mathsf{L}}\left(z\right)$ is holomorphic in $\mathbb{C}\setminus \gamma$. Due to the fact that $Z_n^{\mathsf{L}}\left(z\right)$ has a constant jump on the curve $\gamma$, the matrix function ${\displaystyle {\left(Z_n^{\mathsf{L}}\right)}^{\prime }}$ has the same constant jump on the curve $\gamma$, so the matrix $M_n^{\mathsf{L}}\left(z\right)$ has no jump on the curve $\gamma$, and it follows that, at the origin, $M_n^{\mathsf{L}}\left(z\right)$ has an isolated singularity. From (15) and (12), it holds

$$\begin{array}{cc}\hfill M_n^{\mathsf{L}}\left(z\right)& ={\left(Z_n^{\mathsf{L}}\right)}^{\prime }\left(z\right){\left(Z_n^{\mathsf{L}}\left(z\right)\right)}^{-1}\hfill \\ \hfill & ={\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left(z\right){\left(Y_n^{\mathsf{L}}\left(z\right)\right)}^{-1}+\frac{1}{z}{Y}_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{h}^{\mathsf{L}}\left(z\right)& 0_N\\ 0_N& -h^{\mathsf{R}}\left(z\right)\end{array}\right]{\left(Y_n^{\mathsf{L}}\left(z\right)\right)}^{-1},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill Y_n^{\mathsf{L}}\left(z\right)& =\left[\begin{array}{cc}{P}_n^{\mathsf{L}}\left(z\right)& Q_n^{\mathsf{L}}\left(z\right)\\ -C_{n-1}{P}_{n-1}^{\mathsf{L}}\left(z\right)& -C_{n-1}{Q}_{n-1}^{\mathsf{L}}\left(z\right)\end{array}\right].\hfill \end{array}$$

Each entry of the matrix $Q_n^{\mathsf{L}}\left(z\right)$ is the Cauchy transform of certain function $f\left(z\right)$, where ${\displaystyle f\left(z\right)=\sum _{i\in I}{\varphi }_i\left(z\right)z^{{\alpha }_i}{log}^{p_i}z}$, ${\varphi }_i\left(z\right)$ is an entire function, $Re\left({\alpha }_i\right)>-1$, $p_i\in \mathbb{N}\cup \left\{0\right\}$, and I is a finite set of indices.

It is clear that $\underset{z\to 0}{lim}zf\left(z\right)=0$. Now, (see [56] Sections 8.3–8.6) and [58], its Cauchy transform ${\displaystyle g\left(z\right)=\frac{1}{2\pi i}{\int }_{\gamma }\frac{f\left(t\right)}{t-z}dt}$ also satisfies the same property ${\displaystyle \underset{z\to 0}{lim}zg\left(z\right)=0}$. We can also see that ${\displaystyle \underset{z\to 0}{lim}{z}^2{g}^{\prime }\left(z\right)=0}$. Indeed,

$$\begin{array}{cc}\hfill z{g}^{\prime }\left(z\right)& ={\int }_{\gamma }\frac{zf\left(t\right)}{{(t-z)}^2}dt={\int }_{\gamma }\frac{(z-t)f\left(t\right)}{{(t-z)}^2}dt+{\int }_{\gamma }\frac{tf\left(t\right)}{{(t-z)}^2}dt,\hfill \\ \hfill & =-{\int }_{\gamma }\frac{f\left(t\right)}{t-z}dt-\frac{tf\left(t\right)}{t-z}{|}_{\partial \gamma }+{\int }_{\gamma }\frac{{\left(tf\left(t\right)\right)}^{\prime }}{t-z}dt\hfill \\ \hfill & =-\frac{tf\left(t\right)}{t-z}{|}_{\partial \gamma }+{\int }_{\gamma }\frac{t{f}^{\prime }\left(t\right)}{t-z}dt.\hfill \end{array}$$

From the boundary conditions, the first term is zero and we obtain ${\displaystyle z{g}^{\prime }\left(z\right)={\int }_{\gamma }\frac{t{f}^{\prime }\left(t\right)}{t-z}dt}$, and from the definition of f we obtain that $t{f}^{\prime }\left(t\right)$ is a function in the class of f, which we denote by v and, consequently, ${\displaystyle \underset{z\to 0}{lim}{z}^2{g}^{\prime }\left(z\right)=0}$. From these considerations it follows,

$$\begin{array}{cccccc}\hfill {\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left(z\right)& =\left[\begin{array}{cc}O\left(1\right)& r_1^{\mathsf{L}}\left(z\right)\\ O\left(1\right)& r_2^{\mathsf{L}}\left(z\right)\end{array}\right],\hfill & \hfill {\left(Y_n^{\mathsf{L}}\left(z\right)\right)}^{-1}& =\left[\begin{array}{cc}{r}_3^{\mathsf{L}}\left(z\right)& r_4^{\mathsf{L}}\left(z\right)\\ O\left(1\right)& O\left(1\right)\end{array}\right],\hfill & \hfill z& \to 0,\hfill \end{array}$$

where ${\displaystyle \underset{z\to 0}{lim}{z}^2{r}_i^{\mathsf{L}}\left(z\right)=0_N}$, for $i=1,2$, and ${\displaystyle \underset{z\to 0}{lim}z{r}_i^{\mathsf{R}}\left(z\right)=0_N}$, for $i=3,4$, so it holds that

$$\begin{array}{c}\hfill \underset{z\to 0}{lim}{z}^2{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left(z\right){\left(Y_n^{\mathsf{L}}\right)}^{-1}=\underset{z\to 0}{lim}{z}^2\left[\begin{array}{cc}O\left(1\right)r_1^{\mathsf{L}}\left(z\right)+O\left(1\right)r_3^{\mathsf{L}}\left(z\right)& O\left(1\right)r_1^{\mathsf{L}}\left(z\right)+O\left(1\right)r_4^{\mathsf{L}}\left(z\right)\\ O\left(1\right)r_2^{\mathsf{L}}\left(z\right)+O\left(1\right)r_3^{\mathsf{L}}\left(z\right)& O\left(1\right)r_2^{\mathsf{L}}\left(z\right)+O\left(1\right)r_4^{\mathsf{L}}\left(z\right)\end{array}\right]=0_{2N}.\end{array}$$

Similar considerations leads us to the result that

$$\begin{array}{c}\hfill \underset{z\to 0}{lim}z{Y}_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{h}^{\mathsf{L}}\left(z\right)& 0_N\\ 0_N& -h^{\mathsf{R}}\left(z\right)\end{array}\right]{\left(Y_n^{\mathsf{L}}\left(z\right)\right)}^{-1}=0_{2N},\end{array}$$

so we obtain that ${\displaystyle \underset{z\to 0}{lim}{z}^2{M}_n^{\mathsf{L}}\left(z\right)=0_{2N}}$, and hence the matrix function $M_n^{\mathsf{L}}\left(z\right)$ has at most a simple pole at the point $z=0$. □

3.5. Differential Relations from the Riemann–Hilbert Problem

We are interested in the differential equations fulfilled by the biorthogonal matrix polynomials determined by Laguerre type matrices of weights. Here we use the Riemann–Hilbert problem approach in order to derive these differential relations. We use the notation for the structure matrices

$$\begin{array}{cccc}\hfill {\tilde{M}}_n^{\mathsf{L}}\left(z\right)& =z{M}_n^{\mathsf{L}}\left(z\right),\hfill & \hfill {\tilde{M}}_n^{\mathsf{R}}\left(z\right)& =z{M}_n^{\mathsf{R}}\left(z\right),\hfill \end{array}$$

with ${\tilde{M}}_n^{\mathsf{L}}\left(z\right)$ and ${\tilde{M}}_n^{\mathsf{R}}\left(z\right)$ matrices of entire functions.

Proposition 4

(First order differential equation for the fundamental matrices $Y_n^{\mathsf{L}}\left(z\right)$ and $Y_n^{\mathsf{R}}\left(z\right)$). It holds that

$$\begin{array}{cc}\hfill z{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left(z\right)+Y_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{h}^{\mathsf{L}}\left(z\right)& 0_N\\ 0_N& -h^{\mathsf{R}}\left(z\right)\end{array}\right]& ={\tilde{M}}_n^{\mathsf{L}}\left(z\right)Y_n^{\mathsf{L}}\left(z\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill z{\left(Y_n^{\mathsf{R}}\right)}^{\prime }\left(z\right)+\left[\begin{array}{cc}{h}^{\mathsf{R}}\left(z\right)& 0_N\\ 0_N& -h^{\mathsf{L}}\left(z\right)\end{array}\right]Y_n^{\mathsf{R}}\left(z\right)& =Y_n^{\mathsf{R}}\left(z\right){\tilde{M}}_n^{\mathsf{R}}\left(z\right).\hfill \end{array}$$

(18)

Proof. 

Equations (17) and (18) follow immediately from the definition of the matrices $M_n^{\mathsf{L}}\left(z\right)$ and $M_n^{\mathsf{R}}\left(z\right)$ in (15). □

Corollary 2.

Let $h^{\mathsf{L}}\left(z\right)=A^{\mathsf{L}}z+B^{\mathsf{L}}\mathit{and}{h}^{\mathsf{R}}\left(z\right)=A^{\mathsf{R}}z+B^{\mathsf{R}}$ be two first degree matrix polynomials. The left and right fundamental matrices are given, respectively, by

$$\begin{array}{cc}\hfill M_n^{\mathsf{L}}\left(z\right)& =\frac{1}{z}\left[\begin{array}{cc}{A}^{\mathsf{L}}z+[p_{\mathsf{L},n}^1,A^{\mathsf{L}}]+n{I}_N+B^{\mathsf{L}}& A^{\mathsf{L}}{C}_n^{-1}+C_n^{-1}{A}^{\mathsf{R}}\\ -C_{n-1}{A}^{\mathsf{L}}-A^{\mathsf{R}}{C}_{n-1}& -A^{\mathsf{R}}z+[p_{\mathsf{R},n}^1,A^{\mathsf{R}}]-n{I}_N-B^{\mathsf{R}}\end{array}\right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill M_n^{\mathsf{R}}\left(z\right)& =\frac{1}{z}\left[\begin{array}{cc}{A}^{\mathsf{R}}z-[p_{\mathsf{R},n}^1,A^{\mathsf{R}}]+n{I}_N+B^{\mathsf{R}}& -C_{n-1}{A}^{\mathsf{L}}-A^{\mathsf{R}}{C}_{n-1}\\ A^{\mathsf{L}}{C}_n^{-1}+C_n^{-1}{A}^{\mathsf{R}}& -A^{\mathsf{L}}z-[p_{\mathsf{L},n}^1,A^{\mathsf{L}}]-n{I}_N-B^{\mathsf{L}}\end{array}\right].\hfill \end{array}$$

(20)

Proof. 

By considering (17), Theorem 3, the asymptotic expansion at infinity of the fundamental matrix $Y_n^{\mathsf{L}}$ and (5), the asymptotic behavior of the second kind matrix functions at infinity, and using the identities $p_{\mathsf{R},n}^1=-q_{\mathsf{L},n-1}^1$ and $p_{\mathsf{L},n}^1=-q_{\mathsf{R},n-1}^1$ (19) follows. The relation (16) leads to (20). □

We introduce the $\mathcal{N}$ transform, ${\displaystyle \mathcal{N}\left(F\left(z\right)\right)=F^{\prime }\left(z\right)+\frac{F^2\left(z\right)}{z}}$.

Proposition 5

(Second order differential equation for the fundamental matrices). It holds

$$\begin{array}{cc}\hfill \phantom{o}z{\left(Y_n^{\mathsf{L}}\right)}^{″}+{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left[\begin{array}{cc}2h^{\mathsf{L}}+I_N& 0_N\\ 0_N& -2h^{\mathsf{R}}+I_N\end{array}\right]+Y_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}\mathcal{N}\left(h^{\mathsf{L}}\right)& 0_N\\ 0_N& \mathcal{N}(-h^{\mathsf{R}})\end{array}\right]& =\mathcal{N}\left({\tilde{M}}_n^{\mathsf{L}}\right)Y_n^{\mathsf{L}},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \phantom{o}z{\left(Y_n^{\mathsf{R}}\right)}^{″}+\left[\begin{array}{cc}2h^{\mathsf{R}}+I_N& 0_N\\ 0_N& -2h^{\mathsf{L}}+I_N\end{array}\right]{\left(Y_n^{\mathsf{R}}\right)}^{\prime }+\left[\begin{array}{cc}\mathcal{N}\left(h^{\mathsf{R}}\right)& 0_N\\ 0_N& \mathcal{N}(-h^{\mathsf{L}})\end{array}\right]Y_n{}^{\mathsf{R}}\left(z\right)& =Y_n^{\mathsf{R}}\mathcal{N}\left({\tilde{M}}_n^{\mathsf{R}}\right).\hfill \end{array}$$

(22)

Proof. 

Differentiating in (15), we obtain

$$\begin{array}{c}\hfill {\left(Z_n^{\mathsf{L}}\right)}^{″}{\left(Z_n^{\mathsf{L}}\right)}^{-1}=\frac{{\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }}{z}-\frac{{\tilde{M}}_n^{\mathsf{L}}}{z^2}+\frac{{\left({\tilde{M}}_n^{\mathsf{L}}\right)}^2}{z^2}.\end{array}$$

so that

$$\begin{array}{c}\hfill z{\left(Z_n^{\mathsf{L}}\right)}^{″}{\left(Z_n^{\mathsf{L}}\right)}^{-1}+{\left(Z_n^{\mathsf{L}}\right)}^{\prime }{\left(Z_n^{\mathsf{L}}\right)}^{-1}={\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }+\frac{{\left({\tilde{M}}_n^{\mathsf{L}}\right)}^2}{z}.\end{array}$$

Now, using (12) and (10), we obtain the stated result (21). Equation (22) follows in a similar way from definition of $M_n^{\mathsf{R}}$ in (15). □

We introduce the following ${\mathbb{C}}^{2N\times 2N}$ valued functions

$$\begin{array}{cccc}\hfill {\mathsf{H}}_n^{\mathsf{L}}=\left[\begin{array}{cc}{\mathsf{H}}_{1,1,n}^{\mathsf{L}}& {\mathsf{H}}_{1,2,n}^{\mathsf{L}}\\ {\mathsf{H}}_{2,1,n}^{\mathsf{L}}& {\mathsf{H}}_{2,2,n}^{\mathsf{L}}\end{array}\right]& :=\mathcal{N}\left({\tilde{M}}_n^{\mathsf{L}}\right),\hfill & \hfill {\mathsf{H}}_n^{\mathsf{R}}=\left[\begin{array}{cc}{\mathsf{H}}_{1,1,n}^{\mathsf{R}}& {\mathsf{H}}_{1,2,n}^{\mathsf{R}}\\ {\mathsf{H}}_{2,1,n}^{\mathsf{R}}& {\mathsf{H}}_{2,2,n}^{\mathsf{R}}\end{array}\right]& :=\mathcal{N}\left({\tilde{M}}_n^{\mathsf{R}}\right).\hfill \end{array}$$

It holds that the second order matrix differential Equations (21) and (22) split in the following differential relations

$$\begin{array}{cc}\hfill z{\left(P_n^{\mathsf{L}}\right)}^{″}+{\left(P_n^{\mathsf{L}}\right)}^{\prime }\left(2h^{\mathsf{L}}+I_N\right)+P_n^{\mathsf{L}}\mathcal{N}\left(h^{\mathsf{L}}\right)& ={\mathsf{H}}_{1,1,n}^{\mathsf{L}}{P}_n^{\mathsf{L}}-{\mathsf{H}}_{1,2,n}^{\mathsf{L}}{C}_{n-1}{P}_{n-1}^{\mathsf{L}},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill z{\left(Q_n^{\mathsf{L}}\right)}^{″}+{\left(Q_n^{\mathsf{L}}\right)}^{\prime }\left(-2h^{\mathsf{R}}+I_N\right)+Q_n^{\mathsf{L}}\mathcal{N}(-h^{\mathsf{R}})& ={\mathsf{H}}_{1,1,n}^{\mathsf{L}}{Q}_n^{\mathsf{L}}-{\mathsf{H}}_{1,2,n}^{\mathsf{L}}{C}_{n-1}{Q}_{n-1}^{\mathsf{L}},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill z{\left(P_n^{\mathsf{R}}\right)}^{″}+\left(2h^{\mathsf{R}}+I_N\right){\left(P_n^{\mathsf{R}}\right)}^{\prime }+\mathcal{N}\left(h^{\mathsf{R}}\right)P_n^{\mathsf{R}}& =P_n^{\mathsf{R}}{\mathsf{H}}_{1,1,n}^{\mathsf{R}}-P_{n-1}^{\mathsf{R}}{C}_{n-1}{\mathsf{H}}_{2,1,n}^{\mathsf{R}},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill z{\left(Q_n^{\mathsf{R}}\right)}^{″}+\left(-2h^{\mathsf{L}}+I_N\right){\left(Q_n^{\mathsf{R}}\right)}^{\prime }+\mathcal{N}(-h^{\mathsf{L}})Q_n^{\mathsf{R}}& =Q_n^{\mathsf{R}}{\mathsf{H}}_{1,1,n}^{\mathsf{R}}-Q_{n-1}^{\mathsf{R}}{C}_{n-1}{\mathsf{H}}_{2,1,n}^{\mathsf{R}}.\hfill \end{array}$$

(26)

4. A Class of Laguerre Matrix Weights

Let us consider the matrix weight $W\left(z\right)=e^{A_{1}z}{z}^{\alpha }{e}^{A_{2}z}$, $z\in \mathbb{C}$, defined in $\mathbb{C}\setminus [0,+\infty )$ with support on $\gamma =[0,+\infty )$. Here $\alpha$, $A_1$, $A_2\in {\mathbb{C}}^{N\times N}$ are matrices such that $[\alpha ,A_1]=[\alpha ,A_2]=0_N$, with spectrum $\sigma \left(\alpha \right)$, $Re\left(\sigma \left(\alpha \right)\right)\subset (-1,+\infty )$. This class of weights contains in the Hermitian case some of the cases studied in the literature [6,7,11,59].

For this class of Laguerre weights, we obtain, using analytic arguments, an alternative formula for the residue matrix with the simple pole at $z=0$ of the left fundamental matrix. In a similar manner, we could obtain the result for the right fundamental matrix. Notice that the fundamental matrix is completely determined in the previous section (19), where $A^{\mathsf{L}}$, $A^{\mathsf{R}}$, is substituted, respectively, by $A_1$, $A_2$, and $B^{\mathsf{L}}$, $B^{\mathsf{R}}$ by $\frac{\alpha }{2}$. This alternative formula enables us to make an important simplification in the Equation (21) previously obtained.

Accordingly, we choose

$$\begin{array}{ccc}\hfill W^{\mathsf{L}}\left(z\right)& =e^{A_{1}z}{z}^{\frac{\alpha }{2}},\hfill & \hfill W^{\mathsf{R}}\left(z\right)=z^{\frac{\alpha }{2}}{e}^{A_{2}z}.\end{array}$$

Straightforward calculation shows that $h^{\mathsf{L}}$ and $h^{\mathsf{R}}$ appearing in (11) are given by

$$\begin{array}{ccc}\hfill h^{\mathsf{L}}\left(z\right)& =A_{1}z+\frac{\alpha }{2},\hfill & \hfill h^{\mathsf{R}}\left(z\right)=A_{2}z+\frac{\alpha }{2}.\end{array}$$

Proposition 6.

The structure matrix $M_n^{\mathsf{L}}$ defined in (15) has a simple pole given by the following expression:

(1)

If $Re\left(\sigma \left(\alpha \right)\right)\subset (-1,+\infty )$ and $\sigma \left(\alpha \right)\cap \mathbb{N}=\varnothing$, then

$$\begin{array}{cccc}\hfill M_n^{\mathsf{L}}\left(z\right)& =\frac{1}{z}{F}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}\frac{\alpha }{2}& 0_N\\ 0_N& -\frac{\alpha }{2}\end{array}\right]{\left(F_n^{\mathsf{L}}\left(0\right)\right)}^{-1}+O\left(1\right),\hfill & \hfill z& \to 0,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}\left(0\right)={\widehat{Y}}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]\left[\begin{array}{cc}\begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& 0_{N^{-}}\end{array}& \begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& e^{-i\pi J^{-}}\end{array}\\ \begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& I_{N^{-}}-e^{2i\pi J^{-}}\end{array}& \begin{array}{cc}{e}^{i\pi J^{+}}-e^{-i\pi J^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& 0_{N^{-}}\end{array}\end{array}\right]\end{array}{\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]}^{-1},$$

α has the yielding canonical Jordan form, $\alpha =PJ{P}^{-1}$ with

$$\begin{array}{c}\hfill J=\left[\begin{array}{cc}{J}^{+}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& J^{-}\end{array}\right],\end{array}$$

and $N^{+}$ (respectively, $N^{-}$) being the sum of the algebraic multiplicities associated with eigenvalues with positive (respectively, negative) real parts and in $J^{+}$ (respectively, $J^{-}$), we gather together the Jordan blocks of all eigenvalues with positive (respectively, negative) real parts, and ${\widehat{Y}}_n^{\mathsf{L}}\left(z\right)$ being given by

$$\begin{array}{c}\hfill {\widehat{Y}}_n^{\mathsf{L}}\left(z\right):=Y_n^{\mathsf{L}}\left(z\right){\left[\begin{array}{cc}{z}^{-\alpha }& 0_N\\ I_N-e^{2i\pi \alpha }& z^{\alpha }\end{array}\right]}^{-1}.\end{array}$$

(2)

If $\alpha =m{I}_N,m\in \mathbb{N}$

$$\begin{array}{cccc}\hfill M_n^{\mathsf{L}}\left(z\right)& =\frac{1}{z}{F}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}\frac{m}{2}{I}_N& -\frac{z^m}{2\pi i}{I}_N\\ 0_N& -\frac{m}{2}{I}_N\end{array}\right]{\left(F_n^{\mathsf{L}}\left(0\right)\right)}^{-1}+O\left(1\right),\hfill & \hfill z& \to 0,\hfill \end{array}$$

where $F_n^{\mathsf{L}}\left(0\right)={\widehat{Y}}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}{0}_N& \frac{1}{2\pi i}{I}_N\\ -2\pi i{I}_N& 0_N\end{array}\right]$, with

$$\begin{array}{c}\hfill {\widehat{Y}}_n^{\mathsf{L}}\left(z\right):=Y_n^{\mathsf{L}}\left(z\right){\left[\begin{array}{cc}{(logz)}^{-1}{I}_N& 0_N\\ -2\pi i{I}_N& logz{I}_N\end{array}\right]}^{-1}.\end{array}$$

Remark 1.

In the first case, $F_n^{\mathsf{L}}\left(0\right)$ have a simpler form if $Re\left(\sigma \left(\alpha \right)\right)$ are all positive or all negative

• If $Re\left(\sigma \left(\alpha \right)\right)\subset (0,+\infty )$, then ${\displaystyle F_n^{\mathsf{L}}\left(0\right)=Y_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}{I}_N& 0_N\\ 0_N& e^{i\pi \alpha }-e^{-i\pi \alpha }\end{array}\right]}$.
• If $Re\left(\sigma \left(\alpha \right)\right)\subset (-1,0)$, then ${\displaystyle F_n^{\mathsf{L}}\left(0\right)=\underset{z\to 0}{lim}{Y}_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{0}_N& z^{\alpha }{e}^{-i\pi \alpha }\\ z^{\alpha }\left(I_N-e^{2i\pi \alpha }\right)& e^{i\pi \alpha }-e^{-i\pi \alpha }\end{array}\right]}$.

Proof. 

It can be seen that the matrix function $Z_n^{\mathsf{L}}$ defined by

$$\begin{array}{ccccc}\hfill Z_n^{\mathsf{L}}\left(z\right)=Y_n^{\mathsf{L}}\left(z\right)\mathcal{C}\left(z\right),& \hfill & \hfill \mathrm{where}& \hfill & \hfill \mathcal{C}\left(z\right)=\left[\begin{array}{cc}{W}^{\mathsf{L}}\left(z\right)& 0\\ 0& {\left(W^{\mathsf{R}}\left(z\right)\right)}^{-1}\end{array}\right],\end{array}$$

with $W^{\mathsf{L}}\left(z\right)W^{\mathsf{R}}\left(z\right)=W\left(z\right)$, satisfies the following conditions:

• $Z_n^{\mathsf{L}}$ is holomorphic in $\mathbb{C}\setminus [0,+\infty )$.
• ${\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{+}={\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}{e}^{-i\pi \alpha }& e^{-i\pi \alpha }\\ 0& e^{i\pi \alpha }\end{array}\right]$ over $(0,+\infty )$.

Let us start with the first case: $Re\left(\sigma \left(\alpha \right)\right)\subset (-1,+\infty )$ and $\sigma \left(\alpha \right)\cap \left\{\mathbb{N}\right\}=\varnothing$.

In this case, the constant jump matrix $\left[\begin{array}{cc}{e}^{-i\pi \alpha }& e^{-i\pi \alpha }\\ 0& e^{i\pi \alpha }\end{array}\right]$ can be block diagonalized. For that aim, we consider the matrix

$$\begin{array}{ccccc}\hfill P=\left[\begin{array}{cc}{I}_N& e^{-i\pi \alpha }\\ 0& e^{i\pi \alpha }-e^{-i\pi \alpha }\end{array}\right]& \hfill & \hfill \mathrm{such}\mathrm{that}& \hfill & \hfill \left[\begin{array}{cc}{e}^{-i\pi \alpha }& e^{-i\pi \alpha }\\ 0& e^{i\pi \alpha }\end{array}\right]P=P\left[\begin{array}{cc}{e}^{-i\pi \alpha }& 0\\ 0& e^{i\pi \alpha }\end{array}\right].\end{array}$$

So, over the interval $(0,+\infty )$, we have

$$\begin{array}{c}\hfill {\left(Z_n^{\mathsf{L}}\left(z\right)P\right)}_{+}={\left(Z_n^{\mathsf{L}}\left(z\right)P\right)}_{-}\left[\begin{array}{cc}{e}^{-i\pi \alpha }& 0\\ 0& e^{i\pi \alpha }\end{array}\right].\end{array}$$

For $z\in \mathbb{C}\setminus [0,+\infty )$, let us define the matrix

$$\begin{array}{c}\hfill \psi \left(z\right):=\left[\begin{array}{cc}{z}^{\frac{\alpha }{2}}& 0\\ 0& z^{-\frac{\alpha }{2}}\end{array}\right],\end{array}$$

(27)

which satisfies, over $(0,+\infty )$, the following jump condition

$$\begin{array}{c}\hfill {\left(\psi \left(z\right)\right)}_{+}={\left(\psi \left(z\right)\right)}_{-}\left[\begin{array}{cc}{e}^{-i\pi \alpha }& 0\\ 0& e^{i\pi \alpha }\end{array}\right].\end{array}$$

Consequently, the matrix

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}\left(z\right):=Z_n^{\mathsf{L}}\left(z\right)P{\psi }^{-1}\left(z\right),\end{array}$$

has no jump in the interval $(0,+\infty )$. The matrix function $F_n^{\mathsf{L}}$ has an isolated singularity at the origin which, as we will show now, is a removable singularity, i.e., ${\displaystyle \underset{z\to 0}{lim}z{F}_n^{\mathsf{L}}\left(z\right)=0_{2N}}$. From its definition, we have that

$$\begin{array}{cc}\hfill z{F}_n^{\mathsf{L}}\left(z\right)& =\left[\begin{array}{cc}O\left(z\right)& z{s}_1^{\mathsf{L}}\left(z\right)\\ O\left(z\right)& z{s}_2^{\mathsf{L}}\left(z\right)\end{array}\right]\left[\begin{array}{cc}{e}^{A_{1}z}{z}^{\frac{\alpha }{2}}& 0_N\\ 0_N& e^{-A_{2}z}{z}^{-\frac{\alpha }{2}}\end{array}\right]\left[\begin{array}{cc}{I}_N& e^{-i\pi \alpha }\\ 0_N& e^{i\pi \alpha }-e^{-i\pi \alpha }\end{array}\right]\left[\begin{array}{cc}{z}^{-\frac{\alpha }{2}}& 0\\ 0& z^{\frac{\alpha }{2}}\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{cc}O\left(z\right)& z{s}_1^{\mathsf{L}}\left(z\right)\\ O\left(z\right)& z{s}_2^{\mathsf{L}}\left(z\right)\end{array}\right]\left[\begin{array}{cc}{e}^{A_{1}z}& e^{A_{1}z}{e}^{-i\pi \alpha }{z}^{\alpha }\\ 0_N& e^{-A_{2}z}(e^{i\pi \alpha }-e^{-i\pi \alpha })\end{array}\right],\hfill & \hfill z& \to 0,\hfill \end{array}$$

and as $z{s}_1^{\mathsf{L}}$, $z{s}_2^{\mathsf{L}}\to 0_N$ as $z\to 0$ and $O\left(z\right)z^{\alpha }\to 0_N$, as $z\to 0$ (because the eigenvalues of α are bounded from below by $-1$), we conclude that $z{F}_n^{\mathsf{L}}\left(z\right)\to 0_{2N}$ for $z\to 0$. Hence, $F_n^{\mathsf{L}}\left(z\right)$ is a matrix of entire functions.

Now, we want to compute ${\displaystyle F_n^{\mathsf{L}}\left(0\right)=\underset{z\to 0}{lim}{F}_n^{\mathsf{L}}\left(z\right)}$. For this fact, we will elaborate with respect to the sign of the real part of spectrum of α. Notice that,

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}\left(0\right)=\underset{z\to 0}{lim}{Y}_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{e}^{A_{1}z}& e^{A_{1}z}{e}^{-i\pi \alpha }{z}^{\alpha }\\ 0_N& e^{-A_{2}z}(e^{i\pi \alpha }-e^{-i\pi \alpha })\end{array}\right],\end{array}$$

where the limit of each factor does not need to exist.

We separately compute $F_n^{\mathsf{L}}\left(0\right)$ in the cases, when $Re\left(\sigma \left(\alpha \right)\right)\subset (0,+\infty )$ and when $Re\left(\sigma \left(\alpha \right)\right)\subset (-1,0)$, and then we give $F_n^{\mathsf{L}}\left(0\right)$ in general.

Case$Re\left(\sigma \left(\alpha \right)\right)\subset (0,+\infty )$and$Re\left(\sigma \left(\alpha \right)\right)\cap \left\{\mathbb{N}\right\}=\varnothing$

When the real part of all the eigenvalues of α are strictly positive then each limit exists and

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}\left(0\right)=Y_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}{I}_N& 0_N\\ 0_N& e^{i\pi \alpha }-e^{-i\pi \alpha }\end{array}\right].\end{array}$$

Case$Re\left(\sigma \left(\alpha \right)\right)\subset (-1,0)$and$\sigma \left(\alpha \right)\cap \left\{\mathbb{N}\right\}=\varnothing$.

We cannot proceed as before. However, as the limit exists, if we are able to rewrite

$$\begin{array}{c}\hfill Y_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{e}^{A_{1}z}& e^{A_{1}z}{e}^{-i\pi \alpha }{z}^{\alpha }\\ 0_N& e^{-A_{2}z}(e^{i\pi \alpha }-e^{-i\pi \alpha })\end{array}\right]={\widehat{Y}}_n^{\mathsf{L}}\left(z\right)f\left(z\right),\end{array}$$

in terms of two matrix factors ${\widehat{Y}}_n^{\mathsf{L}}\left(z\right)$ and $f\left(z\right)$, a non singular matrix, with f having a well defined limit for $z\to 0$, also being a non-singular matrix, we can ensure the existence of ${\displaystyle \underset{z\to 0}{lim}{\widehat{Y}}_n^{\mathsf{L}}\left(z\right)}$, and ${\displaystyle F_n^{\mathsf{L}}\left(0\right)=\left(\underset{z\to 0}{lim}{\widehat{Y}}_n^{\mathsf{L}}\left(z\right)\right)\left(\underset{z\to 0}{lim}f\left(z\right)\right)}$. This can be achieved by considering

$$\begin{array}{cc}\hfill {\widehat{Y}}_n^{\mathsf{L}}\left(z\right)& :=Y_n^{\mathsf{L}}\left(z\right){\left[\begin{array}{cc}{z}^{-\alpha }& 0_N\\ I_N-e^{2i\pi \alpha }& z^{\alpha }\end{array}\right]}^{-1},\hfill \\ \hfill f\left(z\right)& :=\left[\begin{array}{cc}{z}^{-\alpha }& 0_N\\ I_N-e^{2i\pi \alpha }& z^{\alpha }\end{array}\right]\left[\begin{array}{cc}{e}^{A_{1}z}& e^{A_{1}z}{e}^{-i\pi \alpha }{z}^{\alpha }\\ 0_N& e^{-A_{2}z}(e^{i\pi \alpha }-e^{-i\pi \alpha })\end{array}\right]\hfill \\ & =\left[\begin{array}{cc}{z}^{-\alpha }{e}^{A_{1}z}& e^{A_{1}z}{e}^{-i\pi \alpha }\\ (I_N-e^{2i\pi \alpha })e^{A_{1}z}& (-e^{A_{1}z}+e^{-A_{2}z})(e^{i\pi \alpha }-e^{-i\pi \alpha })z^{\alpha }\end{array}\right].\hfill \end{array}$$

So that,

$$\begin{array}{cccc}\hfill \underset{z\to 0}{lim}f\left(z\right)& =\left[\begin{array}{cc}{0}_N& e^{-i\pi \alpha }\\ I_N-e^{2i\pi \alpha }& 0_N\end{array}\right],\hfill & \hfill F_n^{\mathsf{L}}\left(0\right)& ={\widehat{Y}}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}{0}_N& e^{-i\pi \alpha }\\ I_N-e^{2i\pi \alpha }& 0_N\end{array}\right].\hfill \end{array}$$

General case$Re\left(\sigma \left(\alpha \right)\right)\subset (-1,+\infty )$and$\sigma \left(\alpha \right)\cap \left\{\mathbb{N}\right\}=\varnothing$.

Recalling the canonical Jordan form, we can write $\alpha =PJ{P}^{-1}$ with

$$\begin{array}{c}\hfill J=\left[\begin{array}{cc}{J}^{+}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& J^{-}\end{array}\right],\end{array}$$

and $N^{+}$ (respectively, $N^{-}$) being the sum of the algebraic multiplicities associated with positive (respectively, negative) eigenvalues and in $J^{+}$ (respectively, $J^{-}$), we gather together the Jordan blocks of all positive (respectively, negative) eigenvalues. Hence,

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{e}^{A_{1}z}& e^{A_{1}z}{e}^{-i\pi \alpha }{z}^{\alpha }\\ 0_N& e^{-A_{2}z}(e^{i\pi \alpha }-e^{-i\pi \alpha })\end{array}\right]=\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]\left[\begin{array}{cc}{e}^{A_{1}z}& e^{{\tilde{A}}_{1}z}{e}^{-i\pi J}{z}^J\\ 0_N& e^{-{\tilde{A}}_{2}z}(e^{i\pi J}-e^{-i\pi J})\end{array}\right]{\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]}^{-1}\end{array}$$

with ${\tilde{A}}_j=P^{-1}{A}_{j}P$, $j=1,2$.

Now, as we did in the previous case, with negative eigenvalues only, we left multiply by the following nonsingular matrix

$$\begin{array}{c}\hfill S\left(z\right):=\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]\left[\begin{array}{cc}\begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& z^{-J^{-}}\end{array}& 0_N\\ \begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& I_{N^{-}}-e^{2i\pi J^{-}}\end{array}& \begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& z^{J^{-}}\end{array}\end{array}\right]{\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]}^{-1},\end{array}$$

to obtain

$$\begin{array}{c}\hfill \left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]\left[\begin{array}{cc}\left[\begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& z^{-J^{-}}\end{array}\right]e^{{\tilde{A}}_{1}z}& \left[\begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& z^{-J^{-}}\end{array}\right]e^{{\tilde{A}}_{1}z}\left[\begin{array}{cc}{e}^{-i\pi J^{+}}{z}^{J^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& e^{-i\pi J^{-}}{z}^{J^{-}}\end{array}\right]\\ \left[\begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& I_{N^{-}}-e^{2i\pi J^{-}}\end{array}\right]e^{{\tilde{A}}_{1}z}& \begin{array}{c}\left[\begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& I_{N^{-}}-e^{2i\pi J^{-}}\end{array}\right]e^{{\tilde{A}}_{1}z}\left[\begin{array}{cc}{e}^{-i\pi J^{+}}{z}^{J^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& e^{-i\pi J^{-}}{z}^{J^{-}}\end{array}\right]\\ +\left[\begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& z^{J^{-}}\end{array}\right]e^{-{\tilde{A}}_{2}z}\left[\begin{array}{cc}{e}^{i\pi J^{+}}-e^{-i\pi J^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& e^{i\pi J^{-}}-e^{-i\pi J_{-}}\end{array}\right]\end{array}\end{array}\right]{\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]}^{-1}\end{array}$$

which for $z\to 0$ has a well defined limit, being a non-singular matrix, given by

$$\begin{array}{c}\hfill \left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]\left[\begin{array}{cc}\begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& 0_{N^{-}}\end{array}& \begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& e^{-i\pi J^{-}}\end{array}\\ \begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& I_{N^{-}}-e^{2i\pi J^{-}}\end{array}& \begin{array}{cc}{e}^{i\pi J^{+}}-e^{-i\pi J^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& 0_{N^{-}}\end{array}\end{array}\right]{\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]}^{-1}.\end{array}$$

Thus,

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}\left(0\right)={\widehat{Y}}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]\left[\begin{array}{cc}\begin{array}{cc}{I}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& 0_{N^{-}}\end{array}& \begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& e^{-i\pi J^{-}}\end{array}\\ \begin{array}{cc}{0}_{N^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& I_{N^{-}}-e^{2i\pi J^{-}}\end{array}& \begin{array}{cc}{e}^{i\pi J^{+}}-e^{-i\pi J^{+}}& 0_{N^{+}\times N^{-}}\\ 0_{N^{-}\times N^{+}}& 0_{N^{-}}\end{array}\end{array}\right]{\left[\begin{array}{cc}P& 0_N\\ 0_N& P\end{array}\right]}^{-1}.\end{array}$$

By definition,

$$\begin{array}{c}\hfill M_n^{\mathsf{L}}={\left(Z_n^{\mathsf{L}}\right)}^{\prime }{\left(Z_n^{\mathsf{L}}\right)}^{-1}={\left(F_n^{\mathsf{L}}\right)}^{\prime }{\left(F_n^{\mathsf{L}}\right)}^{-1}+F_n^{\mathsf{L}}{\psi }^{\prime }{\psi }^{-1}{\left(F_n^{\mathsf{L}}\right)}^{-1},\end{array}$$

as $det{F}_n^{\mathsf{L}}\left(z\right)\ne 0$, we know that ${\left(F_n^{\mathsf{L}}\right)}^{\prime }{\left(F_n^{\mathsf{L}}\right)}^{-1}$ has no singularities, while

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}{\psi }^{\prime }{\psi }^{-1}{\left(F_n^{\mathsf{L}}\right)}^{-1}=\frac{1}{z}{F}_n^{\mathsf{L}}\left[\begin{array}{cc}\frac{\alpha }{2}& 0_N\\ 0_N& -\frac{\alpha }{2}\end{array}\right]{\left(F_n^{\mathsf{L}}\right)}^{-1}.\end{array}$$

Consequently, $M_n^{\mathsf{L}}\left(z\right)$ has a simple pole at the origin with

$$\begin{array}{cccc}\hfill M_n^{\mathsf{L}}\left(z\right)& =\frac{1}{z}{F}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}\frac{\alpha }{2}& 0_N\\ 0_N& -\frac{\alpha }{2}\end{array}\right]{\left(F_n^{\mathsf{L}}\left(0\right)\right)}^{-1}+O\left(1\right),\hfill & \hfill z& \to 0.\hfill \end{array}$$

Let us move to the proof of second case, i.e., $\alpha =m{I}_N$, $m\in \mathbb{N}$.

It can be seen that the matrix function $Z_n^{\mathsf{L}}$ satisfies over $(0,+\infty )$ the following jump condition

$$\begin{array}{c}\hfill {\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{+}={\left(Z_n^{\mathsf{L}}\left(z\right)\right)}_{-}\left[\begin{array}{cc}{(-1)}^m{I}_N& {(-1)}^m{I}_N\\ 0& {(-1)}^m{I}_N\end{array}\right].\end{array}$$

For $z\in \mathbb{C}\setminus [0,+\infty )$, instead of (27), let us define the matrix

$$\begin{array}{c}\hfill \psi \left(z\right):=\left[\begin{array}{cc}{z}^{\frac{m}{2}}{I}_N& -\frac{1}{2\pi i}{z}^{\frac{m}{2}}logz{I}_N\\ 0& z^{-\frac{m}{2}}{I}_N\end{array}\right],\end{array}$$

where we take the branch of the logarithmic function defined in $\mathbb{C}\setminus [0,+\infty )$, which satisfies, over $(0,+\infty )$, the same jump condition

$$\begin{array}{c}\hfill {\left(\psi \left(z\right)\right)}_{+}={\left(\psi \left(z\right)\right)}_{-}\left[\begin{array}{cc}{(-1)}^m{I}_N& {(-1)}^m{I}_N\\ 0& {(-1)}^m{I}_N\end{array}\right].\end{array}$$

Consequently, the matrix

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}\left(z\right):=Z_n^{\mathsf{L}}\left(z\right){\psi }^{-1}\left(z\right)\end{array}$$

has no jump in the interval $(0,+\infty )$. The matrix function $F_n^{\mathsf{L}}$ has an isolated singularity at the origin which, as we will show now, is a removable singularity, i.e.,

$$\begin{array}{cc}\hfill z{F}_n^{\mathsf{L}}\left(z\right)& =\left[\begin{array}{cc}O\left(z\right)& z{s}_1^{\mathsf{L}}\left(z\right)\\ O\left(z\right)& z{s}_2^{\mathsf{L}}\left(z\right)\end{array}\right]\left[\begin{array}{cc}O\left(1\right)& 0_N\\ 0_N& O\left(1\right)\end{array}\right]\left[\begin{array}{cc}O\left(1\right)& O(logz)\\ O\left(1\right)& O\left(1\right)\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{cc}O\left(z\right)+z{s}_1^{\mathsf{L}}\left(z\right)& O(zlogz)+z{s}_1^{\mathsf{L}}\left(z\right)\\ O\left(z\right)+z{s}_2^{\mathsf{L}}\left(z\right)& O(zlogz)+z{s}_2^{\mathsf{L}}\left(z\right)\end{array}\right],\hfill & \hfill z& \to 0,\hfill \end{array}$$

and as $z{s}_1^{\mathsf{L}}$, $z{s}_2^{\mathsf{L}}\to 0_N$ as $z\to 0$, we conclude that $z{F}_n^{\mathsf{L}}\left(z\right)\to 0_{2N}$, as $z\to 0$. Hence, $F_n^{\mathsf{L}}\left(z\right)$ is a matrix of entire functions. To compute $F_n^{\mathsf{L}}\left(0\right)$ we notice that,

$$\begin{array}{c}\hfill F_n^{\mathsf{L}}\left(0\right)=\underset{z\to 0}{lim}{Y}_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{e}^{A_{1}z}& \frac{1}{2\pi i}{z}^{m}logz{e}^{A_{1}z}\\ 0_N& e^{-A_{2}z}\end{array}\right].\end{array}$$

For $m=1,2\cdots$ it holds that $F_n^{\mathsf{L}}\left(0\right)=Y_n^{\mathsf{L}}\left(0\right)$. For $m=0$, the limit of each factor inside the limit does not need to exist. As the limit exists, let us write

$$\begin{array}{c}\hfill Y_n^{\mathsf{L}}\left(z\right)\left[\begin{array}{cc}{e}^{A_{1}z}& \frac{1}{2\pi i}logz{e}^{A_{1}z}\\ 0_N& e^{-A_{2}z}\end{array}\right]={\widehat{Y}}_n^{\mathsf{L}}\left(z\right)f\left(z\right),\end{array}$$

with

$$\begin{array}{cc}\hfill {\widehat{Y}}_n^{\mathsf{L}}\left(z\right)& :=Y_n^{\mathsf{L}}\left(z\right){\left[\begin{array}{cc}{(logz)}^{-1}{I}_N& 0_N\\ -2\pi i{I}_N& logz{I}_N\end{array}\right]}^{-1},\hfill \\ \hfill f\left(z\right)& :=\left[\begin{array}{cc}{(logz)}^{-1}{I}_N& 0_N\\ -2\pi i{I}_N& logz{I}_N\end{array}\right]\left[\begin{array}{cc}{e}^{A_{1}z}& \frac{1}{2\pi i}logz{e}^{A_{1}z}\\ 0_N& e^{-A_{2}z}\end{array}\right]\hfill \\ & =\left[\begin{array}{cc}{(logz)}^{-1}{e}^{A_{1}z}& \frac{1}{2\pi i}{e}^{A_{1}z}\\ -2\pi i{e}^{A_{1}z}& -logz(e^{A_{1}z}-e^{-A_{2}z})\end{array}\right].\hfill \end{array}$$

So that,

$$\begin{array}{cccc}\hfill \underset{z\to 0}{lim}f\left(z\right)& =\left[\begin{array}{cc}{0}_N& \frac{1}{2\pi i}{I}_N\\ -2\pi i{I}_N& 0_N\end{array}\right],\hfill & \hfill F_n^{\mathsf{L}}\left(0\right)& ={\widehat{Y}}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}{0}_N& \frac{1}{2\pi i}{I}_N\\ -2\pi i{I}_N& 0_N\end{array}\right].\hfill \end{array}$$

Using the same kind of reasoning as above, we find that $M_n^{\mathsf{L}}\left(z\right)$ has a simple pole at the origin with

$$\begin{array}{cccc}\hfill M_n^{\mathsf{L}}\left(z\right)& =\frac{1}{z}{F}_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}\frac{m}{2}{I}_N& -\frac{z^m}{2\pi i}{I}_N\\ 0_N& -\frac{m}{2}{I}_N\end{array}\right]{\left(F_n^{\mathsf{L}}\left(0\right)\right)}^{-1}+O\left(1\right),\hfill & \hfill z& \to 0,\hfill \end{array}$$

which ends the proof. □

Proposition 7. 

The structure matrix $M_n^{\mathsf{L}}$ has the yielding expression

$$\begin{array}{c}{M}_n^{\mathsf{L}}\left(z\right)=\frac{1}{z}\left[\begin{array}{cc}{A}_{1}z+[p_{\mathsf{L},n}^1,A_1]+n{I}_N+\frac{\alpha }{2}& A_1{C}_n^{-1}+C_n^{-1}{A}_2\\ -C_{n-1}{A}_1-A_2{C}_{n-1}& -A_{2}z+[p_{\mathsf{R},n}^1,A_2]-n{I}_N-\frac{\alpha }{2}\end{array}\right].\end{array}$$

Proof. 

Substitute $A^{\mathsf{L}}$, $A^{\mathsf{R}}$, respectively, by $A_1$, $A_2$, and $B^{\mathsf{L}}$, $B^{\mathsf{R}}$ by $\frac{\alpha }{2}$ in (19) and (20). □

Proposition 8.

Let α, $A_1$ and $A_2$, such that $[\alpha ,A_1]=[\alpha ,A_2]=0_N$, and the real part of spectrum of α, $\sigma \left(\alpha \right)$, is contained on $(-1,+\infty )$ with $\sigma \left(\alpha \right)\cap \left\{\mathbb{N}\right\}=\varnothing$. If there exists $\lambda \in (0,+\infty )$ such that ${\alpha }^2=\lambda I_N$, or $\alpha =m{I}_N$, for some $m\in \{0,1,2,\dots \}$, then the second order differential equation is simplified to

$$\begin{array}{c}z{\left(Y_n^{\mathsf{L}}\right)}^{″}+{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left[\begin{array}{cc}\alpha +I_N+2A_{1}z& 0_N\\ 0_N& I_N-\alpha -2A_{2}z\end{array}\right]\hfill \\ +Y_n^{\mathsf{L}}\left[\begin{array}{cc}{A}_1+A_1\alpha +{A_1}^{2}z& 0_N\\ 0_N& -A_2+A_2\alpha +{A_2}^{2}z\end{array}\right]\hfill \\ \hfill \phantom{o}=\left[\begin{array}{cc}{A}_1+[p_{\mathsf{L},n}^1,A_1^2]+(n{I}_N+\alpha )A_1+A_1^{2}z& A_1^2{C}_n^{-1}-C_n^{-1}{A}_2^2\\ -C_{n-1}{A}_1^2+A_2^2{C}_{n-1}& -A_2-[p_{\mathsf{R},n}^1,A_2^2]+(n{I}_N+\alpha )A_2+A_2^{2}z\end{array}\right]Y_n^{\mathsf{L}}\left(z\right).\end{array}$$

Proof. 

If we take into account that ${\tilde{M}}_n^{\mathsf{L}}\left(z\right)={\tilde{M}}_n^{\mathsf{L}}\left(0\right)+z{\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)$ and that

$$\begin{array}{c}\mathcal{N}\left({\tilde{M}}_n^{\mathsf{L}}\left(z\right)\right)={\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)+{\left({\tilde{M}}_n^{\mathsf{L}}\left(0\right)\right)}^2\frac{1}{z}+{\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right){\tilde{M}}_n^{\mathsf{L}}\left(0\right)\hfill \\ \hfill +{\tilde{M}}_n^{\mathsf{L}}\left(0\right){\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)+{\left({\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)\right)}^{2}z,\end{array}$$

we find that (21), the second order differential equation that the fundamental matrix satisfies, can be written as

$$\begin{array}{c}z{\left(Y_n^{\mathsf{L}}\right)}^{″}+{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left[\begin{array}{cc}\alpha +I_N+2A_{1}z& 0_N\\ 0_N& I_N-\alpha -2A_{2}z\end{array}\right]\hfill \\ +Y_n^{\mathsf{L}}\left[\begin{array}{cc}{A}_1+\frac{1}{2}{A}_1\alpha +\frac{1}{2}\alpha A_1+z{A_1}^2& 0_N\\ 0_N& -A_2+\frac{1}{2}{A}_2\alpha +\frac{1}{2}\alpha A_2+z{A_2}^2\end{array}\right]+\frac{1}{z}{Y}_n^{\mathsf{L}}\left[\begin{array}{cc}{\left(\frac{\alpha }{2}\right)}^2& 0_N\\ 0_N& {\left(\frac{\alpha }{2}\right)}^2\end{array}\right]\\ \hfill =\left({\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)+{\left({\tilde{M}}_n^{\mathsf{L}}\left(0\right)\right)}^2\frac{1}{z}+{\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right){\tilde{M}}_n^{\mathsf{L}}\left(0\right)+{\tilde{M}}_n^{\mathsf{L}}\left(0\right){\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)+{\left({\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)\right)}^{2}z\right)Y_n^{\mathsf{L}}\left(z\right).\end{array}$$

Under the restriction that the real part of the spectrum of α is contained on $(-1,+\infty )$ and $\sigma \left(\alpha \right)\cap \left\{\mathbb{N}\right\}=\varnothing$, the matrix $M_n^{\mathsf{L}}={\left(Z_n^{\mathsf{L}}\right)}^{\prime }{\left(Z_n^{\mathsf{L}}\right)}^{-1}$ has a pole of order 1 at $z=0$, with residue given by

$$\begin{array}{c}\hfill {\tilde{M}}_n^{\mathsf{L}}\left(0\right)=F_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}\frac{\alpha }{2}& 0_N\\ 0_N& -\frac{\alpha }{2}\end{array}\right]{\left(F_n^{\mathsf{L}}\left(0\right)\right)}^{-1}.\end{array}$$

If we now also assume on the matrix α that ${\alpha }^2=\lambda I_N$, we obtain

$$\begin{array}{c}\hfill {\left({\tilde{M}}_n^{\mathsf{L}}\left(0\right)\right)}^2=F_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}{\left(\frac{\alpha }{2}\right)}^2& 0_N\\ 0_N& {\left(\frac{\alpha }{2}\right)}^2\end{array}\right]{\left(F_n^{\mathsf{L}}\left(0\right)\right)}^{-1}=\frac{\lambda }{4}{I}_{2N}.\end{array}$$

In the case that $\alpha =m{I}_N$, for some $m\in \{0,1,2,\dots \}$, we find that

$$\begin{array}{c}\hfill {\left({\tilde{M}}_n^{\mathsf{L}}\left(0\right)\right)}^2=F_n^{\mathsf{L}}\left(0\right)\left[\begin{array}{cc}{\left(\frac{m}{2}\right)}^2& 0_N\\ 0_N& {\left(\frac{m}{2}\right)}^2\end{array}\right]{\left(F_n^{\mathsf{L}}\left(0\right)\right)}^{-1}=\frac{m^2}{4}{I}_{2N}.\end{array}$$

In both cases, we have

$$\begin{array}{c}z{\left(Y_n^{\mathsf{L}}\right)}^{″}+{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left[\begin{array}{cc}\alpha +I_N+2A_{1}z& 0_N\\ 0_N& I_N-\alpha -2A_{2}z\end{array}\right]\hfill \\ +Y_n^{\mathsf{L}}\left[\begin{array}{cc}{A}_1+A_1\alpha +{A_1}^{2}z& 0_N\\ 0_N& -A_2+A_2\alpha +{A_2}^{2}z\end{array}\right]\\ \hfill =\left({\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)+{\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right){\tilde{M}}_n^{\mathsf{L}}\left(0\right)+{\tilde{M}}_n^{\mathsf{L}}\left(0\right){\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)+{\left({\left({\tilde{M}}_n^{\mathsf{L}}\right)}^{\prime }\left(0\right)\right)}^{2}z\right)Y_n^{\mathsf{L}}\left(z\right).\end{array}$$

and substituting

$$\begin{array}{c}{\tilde{M}}_n^{\mathsf{L}}\left(z\right)=\left[\begin{array}{cc}{A}_{1}z+[p_{\mathsf{L},n}^1,A_1]+n{I}_N+\frac{\alpha }{2}& A_1{C}_n^{-1}+C_n^{-1}{A}_2\\ -C_{n-1}{A}_1-A_2{C}_{n-1}& -A_{2}z+[p_{\mathsf{R},n}^1,A_2]-n{I}_N-\frac{\alpha }{2}\end{array}\right].\end{array}$$

The result follows. □

Remark 2.

If the spectrum of α is contained in $(-1,+\infty )\setminus \mathbb{N}$ then:

• when $\left|\lambda \right|<1$, then the $\pm \lambda$ are admissible eigenvalues for α,
• when$\left|\lambda \right|>1$, then only positive and bigger than 1 eigenvalues are admissible for α, and we have$\alpha =\lambda I_N$.

Corollary 3.

Let us consider$N=1$(i.e., the scalar case). If$A_1=A_2=-\frac{1}{2}$, and$\alpha >-1$, then the second order equation for${\left\{P_n^L\right\}}_{n\in \mathbb{N}}$and${\left\{Q_n^L\right\}}_{n\in \mathbb{N}}$is given by

$$\begin{array}{cc}\hfill z{P}_n^{″}\left(z\right)-(z-\alpha -1)P_n^{\prime }\left(z\right)& =-n{P}_n\left(z\right),\hfill \\ \hfill z{Q}_n^{″}\left(z\right)+(z-\alpha +1)Q_n^{\prime }\left(z\right)& =-(n+1)Q_n\left(z\right).\hfill \end{array}$$

Proof. 

In the scalar case, this equation reduces to

$$\begin{array}{c}z{\left(Y_n^{\mathsf{L}}\right)}^{″}+{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left[\begin{array}{cc}\alpha +1+2A_{1}z& 0\\ 0& 1-\alpha -2A_{1}z\end{array}\right]\hfill \\ +Y_n^{\mathsf{L}}\left[\begin{array}{cc}{A}_1+A_1\alpha +{A_1}^{2}z& 0\\ 0& -A_1+A_1\alpha +{A_1}^{2}z\end{array}\right]\\ \hfill =\left[\begin{array}{cc}{A}_1+(n+\alpha )A_1+A_1^{2}z& 0\\ 0& -A_1+(n+\alpha )A_1+A_1^{2}z\end{array}\right]Y_n^{\mathsf{L}}\left(z\right),\end{array}$$

as $A_1^2{C}_n^{-1}=C_n^{-1}{A}_1^2$ and $A_1=A_2=-\frac{1}{2}$, and so

$$\begin{array}{c}z{\left(Y_n^{\mathsf{L}}\right)}^{″}+{\left(Y_n^{\mathsf{L}}\right)}^{\prime }\left[\begin{array}{cc}\alpha +1-z& 0\\ 0& 1-\alpha +z\end{array}\right]+Y_n^{\mathsf{L}}\left[\begin{array}{cc}-\frac{1}{2}& 0\\ 0& \frac{1}{2}\end{array}\right]=\left[\begin{array}{cc}-\frac{n+1}{2}& 0\\ 0& -\frac{n-1}{2}\end{array}\right]Y_n^{\mathsf{L}}\left(z\right),\end{array}$$

now, considering the $(1,1)$ and the $(1,2)$ entry of this differential matrix equation the result follows. □

5. Matrix Discrete Painlevé IV

We can consider, using the notation introduced before, the matrix weight measure $W=W_{\mathsf{L}}{W}_{\mathsf{R}}$ such that

$$\begin{array}{ccc}\hfill z{\left(W^{\mathsf{L}}\right)}^{\prime }\left(z\right)=(h_0^{\mathsf{L}}+h_1^{\mathsf{L}}z+h_2^{\mathsf{L}}{z}^2)W^{\mathsf{L}}\left(z\right),& \hfill & \hfill z{\left(W^{\mathsf{R}}\right)}^{\prime }\left(z\right)=W^{\mathsf{R}}\left(z\right)(h_0^{\mathsf{R}}+h_1^{\mathsf{R}}z+h_2^{\mathsf{R}}{z}^2).\end{array}$$

From Proposition 5 we find that the matrix

$$\begin{array}{c}\hfill {\tilde{M}}_n=z{M}_n^{\mathsf{L}}\end{array}$$

is given explicitly by

$$\begin{array}{cc}\hfill {\left({\tilde{M}}_n^{\mathsf{L}}\right)}_{11}& =C_n^{-1}{h}_2^{\mathsf{R}}{C}_{n-1}+(h_0^{\mathsf{L}}+h_1^{\mathsf{L}}z+h_2^{\mathsf{L}}{z}^2)+h_1^{\mathsf{L}}{q}_{\mathsf{R},n-1}^1+p_{\mathsf{L},n}^1{h}_1^{\mathsf{L}}\hfill \\ \hfill & \phantom{olaolaolaola}+z(h_2^{\mathsf{L}}{q}_{\mathsf{R},n-1}^1+p_{\mathsf{L},n}^1{h}_2^{\mathsf{L}})+h_2^{\mathsf{L}}{q}_{\mathsf{R},n-1}^2+p_{\mathsf{L},n}^2{h}_2^{\mathsf{L}}+p_{\mathsf{L},n}^1{h}_2^{\mathsf{L}}{q}_{\mathsf{R},n-1}^1+n{I}_N,\hfill \\ \hfill {\left({\tilde{M}}_n^{\mathsf{L}}\right)}_{12}& =(h_1^{\mathsf{L}}+h_2^{\mathsf{L}}z+h_2^{\mathsf{L}}{q}_{\mathsf{R},n}^1+p_{\mathsf{L},n}^1{h}_2^{\mathsf{L}})C_n^{-1}+C_n^{-1}(h_1^{\mathsf{R}}+h_2^{\mathsf{R}}z+h_2^{\mathsf{R}}{p}_{\mathsf{R},n}^1+q_{\mathsf{L},n}^1{h}_2^{\mathsf{R}}),\hfill \\ \hfill {\left({\tilde{M}}_n^{\mathsf{L}}\right)}_{21}& =-C_{n-1}(h_1^{\mathsf{L}}+h_2^{\mathsf{L}}z+h_2^{\mathsf{L}}{q}_{\mathsf{R},n-1}^1+p_{\mathsf{L},n-1}^1{C}_{\mathsf{L}})\hfill \\ \hfill & \phantom{olaolaolaola}-(h_1^{\mathsf{R}}+h_2^{\mathsf{R}}z+h_2^{\mathsf{R}}{p}_{\mathsf{R},n-1}^1+q_{\mathsf{L},n-1}^1{h}_2^{\mathsf{R}})C_{n-1},\hfill \\ \hfill {\left({\tilde{M}}_n^{\mathsf{L}}\right)}_{22}& =-C_{n-1}{h}_2^{\mathsf{L}}{C}_n^{-1}-(h_0^{\mathsf{R}}+h_1^{\mathsf{R}}z+h_2^{\mathsf{R}}{z}^2)-h_1^{\mathsf{R}}{p}_{\mathsf{R},n}^1-q_{\mathsf{L},n-1}^1{h}_1^{\mathsf{R}}\hfill \\ \hfill & \phantom{olaolaolaola}-z(h_2^{\mathsf{R}}{p}_{\mathsf{R},n}^1+q_{\mathsf{L},n-1}^1{h}_2^{\mathsf{R}})-h_2^{\mathsf{R}}{p}_{\mathsf{R},n}^2-q_{\mathsf{L},n-1}^2{h}_2^{\mathsf{R}}-q_{\mathsf{L},n-1}^1{h}_2^{\mathsf{R}}{p}_{\mathsf{R},n}^1-n{I}_N.\hfill \end{array}$$

From the three-term recurrence relation for ${\left\{P_n^{\mathsf{L}}\right\}}_{n\in \mathbb{N}}$ we find that $p_{\mathsf{L},n}^1-p_{\mathsf{L},n+1}^1={\beta }_n^{\mathsf{L}}$ and $p_{\mathsf{L},n}^2-p_{\mathsf{L},n+1}^2={\beta }_n^{\mathsf{L}}{p}_{\mathsf{L},n}^1+{\gamma }_n^{\mathsf{L}}$ where ${\gamma }_n^{\mathsf{L}}=C_n^{-1}{C}_{n-1}$. Consequently,

$$\begin{array}{cccc}\hfill p_{\mathsf{L},n}^1& =-\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},\hfill & \hfill p_{\mathsf{L},n}^2& =\sum _{i,j=0}^{n-1}{\beta }_i^{\mathsf{L}}{\beta }_j^{\mathsf{L}}-\sum _{k=0}^{n-1}{\gamma }_k^{\mathsf{L}}.\hfill \end{array}$$

In the same manner, from the three-term recurrence relation for ${\left\{Q_n^{\mathsf{L}}\right\}}_{n\in \mathbb{N}}$, we deduce that $q_{\mathsf{L},n}^1-q_{\mathsf{L},n-1}^1={\beta }_n^{\mathsf{R}}:=C_n{\beta }_n^{\mathsf{L}}{C}_n^{-1}$ and $q_{\mathsf{L},n}^2-q_{\mathsf{L},n-1}^2={\beta }_n^{\mathsf{R}}{q}_{\mathsf{L},n}^1+{\gamma }_n^{\mathsf{R}}$, where ${\gamma }_n^{\mathsf{R}}=C_n{C}_{n+1}^{-1}$.

If we consider that $W=W^{\mathsf{L}}$ and $W^{\mathsf{R}}=I_N$, and use the representation for ${\left\{P_n^{\mathsf{L}}\right\}}_{n\in \mathbb{N}}$ and ${\left\{Q_n^{\mathsf{L}}\right\}}_{n\in \mathbb{N}}$ in z powers, the $(1,2)$ and $(2,2)$ entries in (17) read

$$\begin{array}{cc}\hfill & (2n+1)I_N+h_0^{\mathsf{L}}+h_2^{\mathsf{L}}({\gamma }_{n+1}^{\mathsf{L}}+{\gamma }_n^{\mathsf{L}}+{\left({\beta }_n^{\mathsf{L}}\right)}^2)+h_1^{\mathsf{L}}{\beta }_n^{\mathsf{L}}\hfill \\ \hfill & \phantom{olaolaolaolaolaolaolaolaolaolaolaola}=[p_{\mathsf{L},n}^1,h_2^{\mathsf{L}}]p_{\mathsf{L},n+1}^1-[p_{\mathsf{L},n}^2,h_2^{\mathsf{L}}]-[p_{\mathsf{L},n}^1,h_1^{\mathsf{L}}],\hfill & \\ \hfill & {\beta }_n^{\mathsf{L}}={\gamma }_n^{\mathsf{L}}\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n-1}^{\mathsf{L}})+[p_{\mathsf{L},n-1}^1,h_2^{\mathsf{L}}]+h_1^{\mathsf{L}}\right)-\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n+1}^{\mathsf{L}})+[p_{\mathsf{L},n}^1,h_2^{\mathsf{L}}]+h_1^{\mathsf{L}}\right){\gamma }_{n+1}^{\mathsf{L}}.\hfill \end{array}$$

We can write these equations as follows

$$\begin{array}{c}(2n+1)I_N+h_0^{\mathsf{L}}+h_2^{\mathsf{L}}({\gamma }_{n+1}^{\mathsf{L}}+{\gamma }_n^{\mathsf{L}}))+(h_2^{\mathsf{L}}{\beta }_n^{\mathsf{L}}+h_1^{\mathsf{L}}){\beta }_n^{\mathsf{L}}\hfill \\ \hfill =\left[\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right]\sum _{k=0}^n{\beta }_k^{\mathsf{L}}-\left[\sum _{i,j=0}^{n-1}{\beta }_i^{\mathsf{L}}{\beta }_j^{\mathsf{L}}-\sum _{k=0}^{n-1}{\gamma }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right]-\left[\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},h_1^{\mathsf{L}}\right],\end{array}$$

(28)

$$\begin{array}{c}{\beta }_n^{\mathsf{L}}-{\gamma }_n^{\mathsf{L}}\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n-1}^{\mathsf{L}})+h_1^{\mathsf{L}}\right)+\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n+1}^{\mathsf{L}})+h_1^{\mathsf{L}}\right){\gamma }_{n+1}^{\mathsf{L}}\hfill \\ \hfill =-{\gamma }_n^{\mathsf{L}}\left[\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right]+\left[-\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right]{\gamma }_{n+1}^{\mathsf{L}}.\end{array}$$

(29)

We will show now that this system contains a noncommutative version of an instance of discrete Painlevé IV equation, as happens in the analogous case for the scalar scenario.

We see, on the right hand side of the nonlinear discrete Equations (28) and (29) nonlocal terms (sums) in the recursion coefficients ${\beta }_n^{\mathsf{L}}$ and ${\gamma }_n^{\mathsf{L}}$, all of them inside commutators. These nonlocal terms vanish whenever the three matrices $\{h_0^{\mathsf{L}},h_1^{\mathsf{L}},h_2^{\mathsf{L}}\}$ conform an Abelian set. Moreover, $\{h_0^{\mathsf{L}},h_1^{\mathsf{L}},h_2^{\mathsf{L}},{\beta }_n^{\mathsf{L}},{\gamma }_n^{\mathsf{L}}\}$ is also an Abelian set. In this commutative setting, we have

$$\begin{array}{c}\hfill (2n+1)I_N+h_0^{\mathsf{L}}+h_2^{\mathsf{L}}({\gamma }_{n+1}^{\mathsf{L}}+{\gamma }_n^{\mathsf{L}}))+(h_2^{\mathsf{L}}{\beta }_n^{\mathsf{L}}+h_1^{\mathsf{L}}){\beta }_n^{\mathsf{L}}=0_N,\\ \hfill {\beta }_n^{\mathsf{L}}-{\gamma }_n^{\mathsf{L}}\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n-1}^{\mathsf{L}})+h_1^{\mathsf{L}}\right)+\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n+1}^{\mathsf{L}})+h_1^{\mathsf{L}}\right){\gamma }_{n+1}^{\mathsf{L}}=0_N.\end{array}$$

In terms of ${\displaystyle {\xi }_n:=\frac{h_0^{\mathsf{L}}}{2}+n{I}_N+h_2^{\mathsf{L}}{\gamma }_n}$ and ${\mu }_n:=h_2^{\mathsf{L}}{\beta }_n^{\mathsf{L}}+h_1^{\mathsf{L}}$ the above equations are

$$\begin{array}{cccc}\hfill {\beta }_n^{\mathsf{L}}{\mu }_n& =-({\xi }_n+{\xi }_{n+1}),\hfill & \hfill {\beta }_n^{\mathsf{L}}({\xi }_n-{\xi }_{n+1})& =-{\gamma }_n{\mu }_{n-1}+{\gamma }_{n+1}{\mu }_{n+1}.\hfill \end{array}$$

Now, we multiply the second equation by ${\displaystyle {\mu }_n}$ and taking into account the first one we arrive at

$$\begin{array}{c}\hfill -({\xi }_n+{\xi }_{n+1})({\xi }_n-{\xi }_{n+1})=-{\gamma }_n{\mu }_{n-1}{\mu }_n+{\gamma }_{n+1}{\mu }_n{\mu }_{n+1}\end{array}$$

and so

$$\begin{array}{c}\hfill {\xi }_{n+1}^2-{\xi }_n^2={\gamma }_{n+1}{\mu }_n{\mu }_{n+1}-{\gamma }_n{\mu }_{n-1}{\mu }_n.\end{array}$$

Hence,

$$\begin{array}{cccccc}\hfill {\xi }_{n+1}^2-{\xi }_0^2={\gamma }_{n+1}{\mu }_n{\mu }_{n+1}& \hfill & \hfill \mathrm{and}& \hfill & \hfill {\beta }_n^{\mathsf{L}}{\mu }_n& =-({\xi }_n+{\xi }_{n+1})\hfill \end{array}$$

(30)

coincide to the ones presented in [60] as a discrete Painlevé IV (dPIV) equation. In fact, taking ${\nu }_n={\mu }_n^{-1}$ we finally arrive at

$$\begin{array}{cccccc}\hfill {\nu }_n{\nu }_{n+1}=\frac{h_2^{\mathsf{L}}\left({\xi }_{n+1}-h_0^{\mathsf{L}}/2-n{I}_N\right)}{{\xi }_{n+1}^2-{\xi }_0^2}& \hfill & \hfill \mathrm{and}& \hfill & \hfill {\xi }_n+{\xi }_{n+1}& =\left({\left(h_2^{\mathsf{L}}\right)}^{-1}{h}_1^{\mathsf{L}}-{\left(h_2^{\mathsf{L}}\right)}^{-1}{\nu }_n^{-1}\right){\nu }_n^{-1}.\hfill \end{array}$$

If we take $h_1^{\mathsf{L}}=0$ in (30), then ${\mu }_n=h_2^{\mathsf{L}}{\beta }_n^{\mathsf{L}}$, and so

$$\begin{array}{c}\hfill {\left({\beta }_n^{\mathsf{L}}\right)}^2{h}_2^{\mathsf{L}}=-({\xi }_n+{\xi }_{n+1}).\end{array}$$

Now, taking square in the first equation in (30) we obtain

$$\begin{array}{c}\hfill \left({\xi }_n+{\xi }_{n+1}\right)\left({\xi }_{n+1}+{\xi }_{n+2}\right)={\left({\left({\xi }_{n+1}-\frac{h_0^{\mathsf{L}}}{2}-n{I}_N\right)}^{-1}\left({\xi }_{n+1}^2-{\xi }_0^2\right)\right)}^2,\end{array}$$

which is an instance of dPIV by Grammaticos, Hietarinta, and Ramani (cf. [61]).

Thus, (28) and (29) for $B_{\mathsf{L}}=0_N$ may be considered as a non-Abelian extension of this instance of dPIV.

We have just seen that,

Theorem 4 (Non-Abelian extension of the dPIV).

When$B_{\mathsf{L}}=0_N$, the following nonlocal nonlinear non-Abelian system for the recursion coefficients is fulfilled

$$\begin{array}{cc}\hfill & (2n+1)I_N+h_0^{\mathsf{L}}+h_2^{\mathsf{L}}({\gamma }_{n+1}^{\mathsf{L}}+{\gamma }_n^{\mathsf{L}}))+h_2^{\mathsf{L}}{\left({\beta }_n^{\mathsf{L}}\right)}^2\hfill \\ \hfill & \phantom{olaolaolaolaolaolaolaola}=\left[\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right]\sum _{k=0}^n{\beta }_k^{\mathsf{L}}-\left[\sum _{i,j=0}^{n-1}{\beta }_i^{\mathsf{L}}{\beta }_j^{\mathsf{L}}-\sum _{k=0}^{n-1}{\gamma }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right],\hfill \\ \hfill & {\beta }_n^{\mathsf{L}}-{\gamma }_n^{\mathsf{L}}\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n-1}^{\mathsf{L}})\right)+\left(h_2^{\mathsf{L}}({\beta }_n^{\mathsf{L}}+{\beta }_{n+1}^{\mathsf{L}})\right){\gamma }_{n+1}^{\mathsf{L}}\hfill \\ \hfill & \phantom{olaolaolaolaolaolaolaola}=-{\gamma }_n^{\mathsf{L}}\left[\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right]+\left[-\sum _{k=0}^{n-1}{\beta }_k^{\mathsf{L}},h_2^{\mathsf{L}}\right]{\gamma }_{n+1}^{\mathsf{L}}.\hfill \end{array}$$

Moreover, this system reduces in the commutative context to the standard dPIV equation.

6. Conclusions and Future Work

In this paper, using the Riemann–Hilbert problem for the Laguerre-type weight matrices, we obtain differential properties of the corresponding matrix biorthogonal polynomials as well as for the second kind matrix functions. It is remarkable to notice that we do not explicitly know the matrix measure, but only its differential properties.

We will consider in future work the case that the support of the measure has two finite end points, the Jacobi-type weight matrices, trying to also obtain differential properties and extensions of Painlevé discrete systems.