An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Abstract

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:${⟨p,q⟩}_s={\int }_{\mathbb{R}}p\left(x\right)q\left(x\right)d\mu \left(x\right)+M_{0}p\left(0\right)q\left(0\right)+M_1{p}^{\prime }\left(0\right)q^{\prime }\left(0\right),$ where $p,q$ are polynomials, $M_0$, $M_1$ are non-negative real numbers and $\mu$ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when $d\mu \left(x\right)=e^{-x^4}dx$.

1. Introduction and Background

Let $\mathbb{P}$ denote the linear space of real polynomials and consider the inner product

$$⟨p,q⟩={\int }_{\mathbb{R}}p\left(x\right)q\left(x\right)d\mu \left(x\right),p,q\in \mathbb{P},$$

(1)

where $\mu$ is a symmetric positive measure supported in some (symmetric with respect to the origin) subset of $\mathbb{R}$. It is easy to see that the associated sequence of monic orthogonal polynomials (MOPS in short) have a symmetry property, i.e., the even degree polynomials are even functions and the odd degree polynomials are odd functions. When $d\mu \left(x\right)=\omega \left(x\right)dx=e^{-V\left(x\right)}dx$, where $V\left(x\right)$ is an even polynomial with positive leading coefficient, the corresponding sequences of polynomials are the so-called Freud type orthogonal polynomials, currently the subject of intense analysis for several choices of the function $V\left(x\right)$ (see, for example [1,2,3,4,5,6,7,8], among many others).

The MOPS is generally denoted by ${\left\{F_n\left(x\right)\right\}}_{n\ge 0}$. We will also use the orthonormal polynomials ${\left\{f_n\right\}}_{n\ge 0}$ satisfying $⟨f_n,f_n⟩=\left|\right|f_n{\left|\right|}^2=1$ for all $n\ge 0$, where the leading coefficient of every orthonormal polynomial is chosen to be positive in order to have uniqueness.

In the present contribution, we consider the following perturbation of (1) that constitutes a diagonal Sobolev-type inner product

$${⟨p,q⟩}_s={\int }_{\mathbb{R}}p\left(x\right)q\left(x\right)d\mu \left(x\right)+M_{0}p\left(0\right)q\left(0\right)+M_1{p}^{\prime }\left(0\right)q^{\prime }\left(0\right),$$

(2)

where $p,q\in \mathbb{P}$, and $M_0$, $M_1$ are nonnegative real numbers. Sobolev inner products have been studied in recent decades because of their applications in approximation theory. Namely, they are used when one wants to obtain a polynomial approximation to both a function and its derivative. For an excellent summary of recent developments on this subject, we refer the reader to the survey [9] and references therein.

We will denote by ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$ its corresponding MOPS and will refer to them as monic Sobolev-type orthogonal polynomials. We will use also the orthonormal version

$$q_n\left(x\right)={\zeta }_n{x}^n+\mathrm{lower}\mathrm{degree}\mathrm{terms},$$

satisfying ${⟨q_n,q_n⟩}_s=\left|\right|q_n{\left|\right|}_s^2=1$. When $d\mu \left(x\right)=e^{-V\left(x\right)},$ (2) is called a Freud–Sobolev type inner product and has also been studied in the literature: the case when $V\left(x\right)=-x^4$ was considered in [10,11], the particular case $M_1=0$ was studied in [12] and the case when $V\left(x\right)=-x^4+2tx$ and $M_1=0$ was analyzed in [13].

In this contribution, we present some general algebraic results for the Sobolev-type orthogonal polynomials, including explicit expressions for the coefficients of a three-term and a five-term recurrence relations satisfied by ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$. We then use these results in the particular case of the Freud–Sobolev type orthogonal polynomials ($V\left(x\right)=e^{-x^4}$) to study some analytic properties of these coefficients. More precisely, we provide asymptotic properties for the confluent version of the Freud kernel polynomial and some of its derivatives. These are later used to deduce the asymptotic behavior for the coefficients of a three-term and a five-term recurrence relations satisfied by ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$. To the best of our knowledge, these results have not be considered elsewhere up to date. For the convenience of the reader, we will repeat some relevant material from [11] without proofs, thus making our exposition self-contained.

The structure of the manuscript is as follows. In Section 2 we put together some connection and recurrence formulas for the monic/orthonormal Sobolev-type polynomials and the orthogonal polynomials, as well as explicit expressions for their coefficients. The main novelty here is a recurrence relation with rational coefficients for the Sobolev-type polynomials (see Theorem 2). Section 3 presents new asymptotic results for the coefficients of the recurrence relations presented in Section 2, in the particular case of the Freud–Sobolev type orthogonal polynomials (see Propositions 3 and 4). Finally, the conclusions and some remarks are outlined in Section 4.

2. Connection and Recurrence Formulas

Let ${\left\{F_n\right\}}_{n\ge 0}$ (resp. ${\left\{f_n\left(x\right)\right\}}_{n\ge 0}$) be the sequence of monic orthogonal (resp. orthonormal) polynomials associated with (1), i.e.,

$$f_n\left(x\right)={\gamma }_n{F}_n\left(x\right)={\gamma }_n{x}^n+\mathrm{lower}\mathrm{degree}\mathrm{terms},$$

where

$${\gamma }_n={\left(\left|\right|F_n{\left|\right|}^2\right)}^{-1/2}>0,$$

(3)

and

$$\left|\right|F_n{\left|\right|}^2={\int }_{\mathbb{R}}{\left[F_n\left(x\right)\right]}^{2}d\mu \left(x\right).$$

It is well-known that they satisfy the three-term recurrence relation

$$x{f}_{n-1}\left(x\right)=a_n{f}_n\left(x\right)+a_{n-1}{f}_{n-2}\left(x\right),n\ge 1,$$

(4)

with $f_{-1}:=0$, $f_0\left(x\right)={\left({\int }_{\mathbb{R}}\omega \left(x\right)dx\right)}^{-1/2}$, $f_1\left(x\right)=a_1^{-1}x$, and $a_n>0$ for $n\ge 1$. For the monic case, we have

$$x{F}_n\left(x\right)=F_{n+1}\left(x\right)+a_n^2{F}_{n-1}\left(x\right),n\ge 1.$$

(5)

The n-th degree reproducing kernel associated with ${\left\{F_n\right\}}_{n⩾0}$ is defined by

$$K_n(x,y)=\sum _{k=0}^n\frac{F_k\left(x\right)F_k\left(y\right)}{\left|\right|F_k{\left|\right|}^2}=\frac{1}{\left|\right|F_n{\left|\right|}^2}\frac{F_{n+1}\left(x\right)F_n\left(y\right)-F_{n+1}\left(y\right)F_n\left(x\right)}{x-y},$$

(6)

where the latter expression is called the Christoffel–Darboux formula and holds for $x\ne y$. Moreover, we have the confluent expression

$$K_n(x,x)=\sum _{k=0}^n\frac{{\left[F_k\left(x\right)\right]}^2}{\left|\right|F_k{\left|\right|}^2}=\frac{{\left[F_{n+1}\right]}^{\prime }\left(x\right)F_n\left(x\right)-{\left[F_n\right]}^{\prime }\left(x\right)F_{n+1}\left(x\right)}{\left|\right|F_n{\left|\right|}^2}.$$

(7)

By using the following standard notation for the partial derivatives

$$\frac{{\partial }^{j+k}{K}_n\left(x,y\right)}{{\partial }^{j}x{\partial }^{k}y}=:K_n^{(j,k)}\left(x,y\right),0\le j,k\le n,$$

(8)

We easily have

$$\begin{array}{ccc}\hfill K_{2n-1}(0,0)& =& \frac{-{\left[F_{2n-1}\right]}^{\prime }\left(0\right)F_{2n}\left(0\right)}{\left|\right|F_{2n-1}{\left|\right|}^2},\hfill \\ \hfill K_{2n-1}^{(1,1)}(0,0)& =& \frac{1}{\left|\right|F_{2n-1}{\left|\right|}^2}\left[\frac{{\left[F_{2n}\right]}^{″}\left(0\right){\left[F_{2n-1}\right]}^{\prime }\left(0\right)}{2}-\frac{{{\left[F_{2n-1}\right]}^{″}}^{\prime }\left(0\right)F_{2n}\left(0\right)}{6}\right],\hfill \end{array}$$

as well as

$$\begin{array}{ccc}\hfill K_{2n}(0,0)& =& \frac{{\left[F_{2n+1}\right]}^{\prime }\left(0\right)F_{2n}\left(0\right)}{\left|\right|F_{2n}{\left|\right|}^2},\hfill \\ \hfill K_{2n}^{(1,1)}(0,0)& =& \frac{1}{\left|\right|F_{2n}{\left|\right|}^2}\left[\frac{{{\left[F_{2n+1}\right]}^{″}}^{\prime }\left(0\right)F_{2n}\left(0\right)}{6}-\frac{{\left[F_{2n}\right]}^{″}\left(0\right){\left[F_{2n+1}\right]}^{\prime }\left(0\right)}{2}\right],\hfill \end{array}$$

In addition, due to the symmetry of ${\left\{F_n\left(x\right)\right\}}_{n\ge 0}$, it is clear that $K_n^{(0,1)}(0,0)=K_n^{(1,0)}(0,0)=0$, $K_{2n+1}(x,0)=K_{2n}(x,0)$, $K_{2n}^{(0,1)}(x,0)=K_{2n-1}^{(0,1)}(x,0)$, and $K_{2n}^{(1,1)}(x,0)=K_{2n-1}^{(1,1)}(x,0)$ for $n\ge 1$.

2.1. Connection Formula

The connection formula relating $Q_n\left(x\right)$ and $F_n\left(x\right)$ for the even and odd cases are

$$\begin{array}{ccc}\hfill Q_{2n}\left(x\right)& =& F_{2n}\left(x\right)-M_0\frac{F_{2n}\left(0\right)}{[1+M_0{K}_{2n-2}(0,0)]}{K}_{2n-2}(x,0),n\ge 1,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill Q_{2n+1}\left(x\right)& =& F_{2n+1}\left(x\right)-M_1\frac{{\left[F_{2n+1}\right]}^{\prime }\left(0\right)}{1+M_1{K}_{2n-1}^{(1,1)}(0,0)}{K}_{2n-1}^{(0,1)}(x,0),n\ge 1,\hfill \end{array}$$

(10)

and were proved in [11] for the particular case $d\mu \left(x\right)=e^{-x^4}dx$, although they are clearly valid for the more general case of any symmetric measure $\mu$. This means that $Q_{2n}$ ($Q_{2n+1}$) is an even (odd) polynomial. In other words, the Sobolev-type perturbation defined in (2) induces two new symmetric orthogonal sequences, associated with the even and odd degree polynomials, respectively. As a consequence, we have

$$\begin{array}{cccccc}\hfill Q_{2n}\left(0\right)& & ={\displaystyle \frac{F_{2n}\left(0\right)}{[1+M_0{K}_{2n-2}(0,0)]}},\hfill & \hfill & \hfill Q_{2n+1}\left(0\right)& =0,\hfill \\ \hfill [Q_{2n+1}{]}^{\prime }\left(0\right)& & ={\displaystyle \frac{{\left[F_{2n+1}\right]}^{\prime }\left(0\right)}{1+M_1{K}_{2n-1}^{(1,1)}(0,0)}},\hfill & \hfill & \hfill {\left[Q_{2n}\right]}^{\prime }\left(0\right)& =0.\hfill \end{array}$$

(11)

Denoting $q_n={\zeta }_n{x}^n+\cdots$ and $f_n={\gamma }_n{x}^n+\cdots$, the orthonormal version of the connection formulas becomes

$$\begin{array}{ccc}\hfill q_{2n}\left(x\right)& =& \frac{{\zeta }_{2n}}{{\gamma }_{2n}}{f}_{2n}\left(x\right)-M_0{q}_{2n}\left(0\right)K_{2n-2}(x,0),n\ge 1\hfill \\ \hfill q_{2n+1}\left(x\right)& =& \frac{{\zeta }_{2n+1}}{{\gamma }_{2n+1}}{f}_{2n+1}\left(x\right)-M_1{\left[q_{2n+1}\right]}^{\prime }\left(0\right)K_{2n-1}^{(0,1)}(x,0),n\ge 0,\hfill \end{array}$$

with

$$q_{2n}\left(0\right)=\frac{{\zeta }_{2n}}{{\gamma }_{2n}}\frac{f_{2n}\left(0\right)}{\left[1+M_0{K}_{2n-2}(0,0)\right]},{\left[q_{2n+1}\right]}^{\prime }\left(0\right)=\frac{{\zeta }_{2n+1}}{{\gamma }_{2n+1}}\frac{{\left[f_{2n+1}\right]}^{\prime }\left(0\right)}{\left[1+M_1{K}_{2n-1}^{(1,1)}(0,0)\right]}.$$

(12)

Notice that we also have the connection formula (see [11])

$$x^2{Q}_n\left(x\right)={\mathcal{A}}_1(n;x)F_n\left(x\right)+{\mathcal{B}}_1(n;x)F_{n-1}\left(x\right),$$

(13)

where ${\mathcal{A}}_1(n;x)=x^2+{\mathcal{A}}_{10}\left(n\right)$, and ${\mathcal{B}}_1(n;x)={\mathcal{B}}_{11}\left(n\right)x$ with

$${\mathcal{A}}_{10}\left(n\right)=-\frac{M_1{\left[Q_n\right]}^{\prime }\left(0\right)F_{n-1}\left(0\right)}{\left|\right|F_{n-1}{\left|\right|}^2},{\mathcal{B}}_{11}\left(n\right)=\frac{M_0{Q}_n\left(0\right)F_n\left(0\right)+M_1{\left[Q_n\right]}^{\prime }\left(0\right){\left[F_n\right]}^{\prime }\left(0\right)}{\left|\right|F_{n-1}{\left|\right|}^2},$$

Notice that the previous connection formulas and expressions for the kernels and their derivatives appear in [11]. On the other hand, the relation between the leading coefficients ${\zeta }_n$ and ${\gamma }_n$, which gives a relation between the corresponding norms, is given in the next result. It also appears in [11] (Prop. 3) in a slightly different form, but we give a different proof here. It will be used in the following section to compute the asymptotic behavior of the ration of the norms. The results appearing in the remainder of the manuscript are new and are not contained in [11].

Proposition 1.

For $n\ge 1$, we have

$$\frac{\parallel F_n{\parallel }^2}{\parallel Q_n{\parallel }_s^2}=\frac{{\zeta }_n^2}{{\gamma }_n^2}={\left(\frac{1+M_0{K}_n(0,0)}{1+M_0{K}_{n-1}(0,0)}+\frac{1+M_1{K}_n^{(1,1)}(0,0)}{1+M_1{K}_{n-1}^{(1,1)}(0,0)}-1\right)}^{-1}.$$

(14)

Moreover,

$$\begin{array}{ccc}\hfill \frac{{\zeta }_{2n}^2}{{\gamma }_{2n}^2}& =& \frac{\parallel F_{2n}{\parallel }^2}{\parallel Q_{2n}{\parallel }_s^2}=\frac{1+M_0{K}_{2n-2}(0,0)}{1+M_0{K}_{2n}(0,0)},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \frac{{\zeta }_{2n+1}^2}{{\gamma }_{2n+1}^2}& =& \frac{\parallel F_{2n+1}{\parallel }^2}{\parallel Q_{2n+1}{\parallel }_s^2}=\frac{1+M_1{K}_{2n-1}^{(1,1)}(0,0)}{1+M_1{K}_{2n+1}^{(1,1)}(0,0)}.\hfill \end{array}$$

(16)

Proof. 

Consider the Fourier expansion

$$q_n\left(x\right)=\sum _{k=0}^n{c}_{k,n}{f}_k\left(x\right),$$

whose coefficients are (see (2))

$$c_{k,n}={\int }_{\mathbb{R}}{q}_n\left(x\right)f_k\left(x\right)d\mu \left(x\right)={⟨q_n\left(x\right),f_k\left(x\right)⟩}_s-M_0{q}_n\left(0\right)f_k\left(0\right)-M_1{\left[q_n\right]}^{\prime }\left(0\right){\left[f_k\right]}^{\prime }\left(0\right).$$

If $k=n$, by comparing the leading coefficients, we obtain $c_{n,n}={\zeta }_n/{\gamma }_n$. When $k<n$, by orthogonality we have ${⟨q_n\left(x\right),f_k\left(x\right)⟩}_s=0$, so that

$$c_{k,n}=-M_0{q}_n\left(0\right)f_k\left(0\right)-M_1{\left[q_n\right]}^{\prime }\left(0\right){\left[f_k\right]}^{\prime }\left(0\right).$$

Hence,

$$\begin{array}{ccc}\hfill {\int }_{\mathbb{R}}{\left[q_n\left(x\right)\right]}^{2}d\mu \left(x\right)& =& {\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2+\sum _{k=0}^{n-1}{\left[c_{k,n}\right]}^2{\left[f_k\left(x\right)\right]}^2\hfill \\ & =& {\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2+M_0^2{q}_n^2\left(0\right)\sum _{k=0}^{n-1}{f}_k^2\left(0\right)+M_1^2{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2\sum _{k=0}^{n-1}{\left({\left[f_k\right]}^{\prime }\left(0\right)\right)}^2\hfill \\ & & +2M_0{M}_1{q}_n\left(0\right){\left[q_n\right]}^{\prime }\left(0\right)\sum _{k=0}^{n-1}{f}_k\left(0\right){\left[f_k\right]}^{\prime }\left(0\right).\hfill \end{array}$$

On the other hand, by the orthonormality of $q_n\left(x\right)$ with respect to (2),

$${⟨q_n,q_n⟩}_s=1={\int }_{\mathbb{R}}{\left[q_n\left(x\right)\right]}^{2}d\mu \left(x\right)+M_0{q}_n^2\left(0\right)+M_1{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2,$$

so that

$${\int }_{\mathbb{R}}{\left[q_n\left(x\right)\right]}^{2}d\mu \left(x\right)=1-M_0{q}_n^2\left(0\right)-M_1{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2.$$

Therefore,

$$1-M_0{q}_n^2\left(0\right)-M_1{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2={\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2+M_0^2{q}_n^2\left(0\right)\sum _{k=0}^{n-1}{f}_k^2\left(0\right)$$

$$+M_1^2{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2\sum _{k=0}^{n-1}{\left({\left[f_k\right]}^{\prime }\left(0\right)\right)}^2+2M_0{M}_1{q}_n\left(0\right){\left[q_n\right]}^{\prime }\left(0\right)\sum _{k=0}^{n-1}{f}_k\left(0\right){\left[f_k\right]}^{\prime }\left(0\right).$$

Taking into account that ${\sum }_{k=0}^{n-1}{f}_k\left(0\right){\left[f_k\right]}^{\prime }\left(0\right)=K_{n-1}^{(0,1)}(0,0)=0$, we rewrite the above expression as

$$1-M_0{q}_n^2\left(0\right)-M_1{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2={\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2+M_0^2{q}_n^2\left(0\right)K_{n-1}(0,0)+M_1^2{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2{K}_{n-1}^{(1,1)}(0,0),$$

and, as a consequence,

$$1={\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2+M_0{q}_n^2\left(0\right)\left[1+M_0{K}_{n-1}(0,0)\right]+M_1{\left({\left[q_n\right]}^{\prime }\left(0\right)\right)}^2\left[1+M_1{K}_{n-1}^{(1,1)}(0,0)\right].$$

Next, using (12) we obtain

$$1={\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2\left[1+M_0\frac{{\left(f_n\left(0\right)\right)}^2}{\left[1+M_0{K}_{n-1}(0,0)\right]}+M_1\frac{{\left({\left[f_n\right]}^{\prime }\left(0\right)\right)}^2}{\left[1+M_1{K}_{n-1}^{(1,1)}(0,0)\right]}\right].$$

Since

$$\begin{array}{ccc}\hfill K_n(0,0)-K_{n-1}(0,0)& =& {\left(f_n\left(0\right)\right)}^2,\hfill \\ \hfill K_n^{(1,1)}(0,0)-K_{n-1}^{(1,1)}(0,0)& =& {\left({\left[f_n\right]}^{\prime }\left(0\right)\right)}^2,\hfill \end{array}$$

we get

$$\begin{array}{ccc}\hfill 1& =& {\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2\left[1+\frac{M_0{K}_n(0,0)-M_0{K}_{n-1}(0,0)}{\left[1+M_0{K}_{n-1}(0,0)\right]}+\frac{M_1{K}_n^{(1,1)}(0,0)-M_1{K}_{n-1}^{(1,1)}(0,0)}{1+M_1{K}_{n-1}^{(1,1)}(0,0)}\right],\hfill \\ \hfill 1& =& {\left(\frac{{\zeta }_n}{{\gamma }_n}\right)}^2\left[\frac{1+M_0{K}_n(0,0)}{1+M_0{K}_{n-1}(0,0)}+\frac{1+M_1{K}_n^{(1,1)}(0,0)}{1+M_1{K}_{n-1}^{(1,1)}(0,0)}-1\right].\hfill \end{array}$$

which is (14)–(16) follow from considering the even and odd degree cases. □

2.2. Five-Term Recurrence Relation

This section deals with the five-term recurrence relation that the sequence ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$ satisfies. We will use the fact that the multiplication operator by $x^2$ is a symmetric operator with respect to (2), i.e.,

$${⟨x^{2}h\left(x\right),g\left(x\right)⟩}_s={⟨h\left(x\right),x^{2}g\left(x\right)⟩}_s.$$

Notice that we also have

$${⟨x^{2}h\left(x\right),g\left(x\right)⟩}_s=⟨x^{2}h\left(x\right),g\left(x\right)⟩.$$

(17)

We need a preliminary result.

Lemma 1.

For every $n\ge 1$, the connection formula

$$\begin{array}{ccc}\hfill x^2{Q}_n\left(x\right)& =& F_{n+2}\left(x\right)+\left[a_{n+1}^2+a_n^2+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right)\right]F_n\left(x\right)\hfill \\ & & +a_{n-1}^2\left[a_n^2+{\mathcal{B}}_{11}\left(n\right)\right]F_{n-2}\left(x\right)\hfill \end{array}$$

holds.

Proof. 

The result follows easily from (13) after successive applications of (5). □

We are ready to find the five-term recurrence relation satisfied by ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$.

Theorem 1 (Five-term recurrence relation).

For every $n\ge 1$, the monic Sobolev type polynomials ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$, orthogonal with respect to (2), satisfy the following five-term recurrence relation

$$x^2{Q}_n\left(x\right)=Q_{n+2}\left(x\right)+{\lambda }_{n,n}{Q}_n\left(x\right)+{\lambda }_{n,n-2}{Q}_{n-2}\left(x\right),n\ge 1,$$

(18)

with initial conditions $Q_{-1}\left(x\right)=0,$$Q_0\left(x\right)=1$, $Q_1\left(x\right)=x$, and $Q_2\left(x\right)=x^2-{\lambda }_{0,0}$, where

$$\begin{array}{ccc}\hfill {\lambda }_{n,n}& =& \left[a_{n+1}^2+a_n^2+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right)\right]\frac{\left|\right|F_n{\left|\right|}^2}{\left|\right|Q_n{\left|\right|}_s^2}+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right),n\ge 0,\hfill \\ \hfill {\lambda }_{n,n-2}& =& a_{n-1}^2\left[a_n^2+{\mathcal{B}}_{11}\left(n\right)\right]\frac{\left|\right|F_{n-2}{\left|\right|}^2}{\left|\right|Q_{n-2}{\left|\right|}_s^2},n\ge 2.\hfill \end{array}$$

Proof. 

Let us consider the Fourier expansion of $x^2{Q}_n\left(x\right)$ in terms of ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$

$$x^2{Q}_n\left(x\right)=\sum _{k=0}^{n+2}{\lambda }_{n,k}{Q}_k\left(x\right),$$

where

$${\lambda }_{n,k}=\frac{{⟨x^2{Q}_n\left(x\right),Q_k\left(x\right)⟩}_s}{\left|\right|Q_k{\left|\right|}_s^2},k=0,\dots ,n+2.$$

(19)

Thus, ${\lambda }_{n,k}=0$ for $k=0,\dots ,n-3$, and ${\lambda }_{n,n+2}=1$. To obtain ${\lambda }_{n,n+1}$, from (13) we get

$$\begin{array}{ccc}\hfill {\lambda }_{n,n+1}& =& \frac{1}{\left|\right|Q_{n+1}{\left|\right|}_s^2}{⟨{\mathcal{A}}_1(n;x)F_n^{\alpha }\left(x\right),Q_{n+1}\left(x\right)⟩}_s+\frac{1}{\left|\right|Q_{n+1}{\left|\right|}_s^2}{⟨{\mathcal{B}}_1(n;x)F_{n-1}\left(x\right),Q_{n+1}\left(x\right)⟩}_s\hfill \\ & =& \frac{1}{\left|\right|Q_{n+1}{\left|\right|}_s^2}{⟨x^2{F}_n\left(x\right),Q_{n+1}\left(x\right)⟩}_s=\frac{1}{\left|\right|Q_{n+1}{\left|\right|}_s^2}⟨F_n\left(x\right),x^2{Q}_{n+1}\left(x\right)⟩=0,\hfill \end{array}$$

by using Lemma 1. In order to compute ${\lambda }_{n,n}$, using again (13) we get

$${\lambda }_{n,n}=\frac{{⟨x^2{F}_n\left(x\right),Q_n\left(x\right)⟩}_s}{\left|\right|Q_n{\left|\right|}_s^2}+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right).$$

But, according to Lemma 1, the first term is

$$\frac{{⟨x^2{F}_n\left(x\right),Q_n\left(x\right)⟩}_1}{\left|\right|Q_n{\left|\right|}_1^2}=\left[a_{n+1}^2+a_n^2+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right)\right]\frac{\left|\right|F_n{\left|\right|}^2}{\left|\right|Q_n{\left|\right|}_s^2},$$

so that

$${\lambda }_{n,n}=\left[a_{n+1}^2+a_n^2+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right)\right]\frac{\left|\right|F_n{\left|\right|}^2}{\left|\right|Q_n{\left|\right|}_s^2}+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right).$$

A similar analysis yields ${\lambda }_{n,n-1}=0$ and

$$\begin{array}{ccc}\hfill {\lambda }_{n,n-2}& =& \frac{⟨x^2{Q}_n\left(x\right),Q_{n-2}\left(x\right)⟩}{\left|\right|Q_{n-2}{\left|\right|}_s^2}=\frac{⟨a_{n-1}^2\left[a_n^2+{\mathcal{B}}_{11}\left(n\right)\right]F_{n-2}\left(x\right),Q_{n-2}\left(x\right)⟩}{\left|\right|Q_{n-2}{\left|\right|}_s^2}\hfill \\ & =& a_{n-1}^2\left[a_n^2+{\mathcal{B}}_{11}\left(n\right)\right]\frac{\left|\right|F_{n-2}{\left|\right|}^2}{\left|\right|Q_{n-2}{\left|\right|}_s^2}.\hfill \end{array}$$

□

2.3. A Recurrence Formula with Rational Coefficients

Now we derive an alternative fundamental recurrence formula but now with rational coefficients, which can be used to obtain the polynomial $Q_{n+1}\left(x\right)$, from the two previous consecutive polynomials $Q_n\left(x\right)$ and $Q_{n-1}\left(x\right)$ in the Sobolev-type SMOP ${\left\{Q_n\right\}}_{n\ge 0}$. This technique has also been implemented in [14], and we will use the results there to obtain the recurrence formula here and the asymptotic behavior of the aforementioned rational coefficients in the next section. For the convenience of the reader, we repeat a few relevant computations from [14] without proofs, thus making our exposition self-contained. We begin with the ladder differential equations for this SMOP, provided in [11] (p. 523) in the compact way

$$\begin{array}{ccc}\hfill (\mathsf{\Xi }(x;n,2)I-D_x)Q_n\left(x\right)& =& \mathsf{\Xi }(x;n,1)Q_{n-1}\left(x\right),\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill (\mathsf{\Theta }(x;n,1)I+D_x)Q_{n-1}\left(x\right)& =& \mathsf{\Theta }(x;n,2)Q_n\left(x\right),\hfill \end{array}$$

(21)

where I and $D_x$ are the identity and x-derivative operators respectively, and

$$\begin{array}{ccc}\hfill \mathsf{\Xi }(x;n,k)& =& \frac{1}{\Lambda (x;n)}\left|\begin{array}{cc}{C}_1(x;n)& A_k(x;n)\\ D_1(x;n)& B_k(x;n)\end{array}\right|,\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill \mathsf{\Theta }(x;n,k)& =& \frac{1}{\Lambda (x;n)}\left|\begin{array}{cc}{C}_2(x;n)& A_k(x;n)\\ D_2(x;n)& B_k(x;n)\end{array}\right|,\hfill \end{array}$$

(23)

for $k=1,2$, where $A_1(x;n):=A_n\left(x\right)$ and $B_1(x;n):=B_n\left(x\right)$.

Shifting the index $n\to n+1$ in (21) and adding the resulting equation to (20) yields

$$(\mathsf{\Xi }(x;n,2)+\Theta (x;n+1,1))Q_n\left(x\right)=\mathsf{\Xi }(x;n,1)Q_{n-1}\left(x\right)+\Theta (x;n+1,2)Q_{n+1}\left(x\right),$$

and now, rearranging the terms in the previous equation, we obtain the desired three term recurrence relation

$$Q_{n+1}\left(x\right)=\tilde{\beta }(x;n)Q_n\left(x\right)+\tilde{\gamma }(x;n)Q_{n-1}\left(x\right),$$

(24)

with rational coefficients

$$\tilde{\beta }(x;n)=\frac{\mathsf{\Xi }(x;n,2)+\Theta (x;n+1,1)}{\Theta (x;n+1,2)},\mathrm{and}\tilde{\gamma }(x;n)=\frac{-\mathsf{\Xi }(x;n,1)}{\Theta (x;n+1,2)}.$$

(25)

In what follows, we rewrite (5) in the same fashion as (24), namely

$$F_{n+1}\left(x\right)=\beta (x;n)F_n\left(x\right)+\gamma (x;n)F_{n-1}\left(x\right),$$

(26)

with $\beta (x;n)=x$, and $\gamma (x;n)=-a_n^2$.

We are now interested in simplifying the above expressions for $\tilde{\beta }(x;n)$ and $\tilde{\gamma }(x;n)$. A trivial verification, already done in Section 2 in [14], shows that the above coefficients only depend on $A_n\left(x\right)$, $B_n\left(x\right)$, $\beta (x;n)$, and $\gamma (x;n)$ as follows

$$\tilde{\beta }(x;n)=\frac{A_{n+1}\left(x\right)B_{n-1}\left(x\right)\gamma (x;n)}{\mathfrak{D}(x;n)}$$

(27)

$$+\frac{\left[A_{n+1}\left(x\right)\beta (x;n)+B_{n+1}\left(x\right)\right]·\left[B_{n-1}\left(x\right)\beta (x;n-1)-A_{n-1}\left(x\right)\gamma (x;n-1)\right]}{\mathfrak{D}(x;n)}$$

and

$$\tilde{\gamma }(x;n)=\frac{\gamma (x;n-1)\left\{B_n\left(x\right)\left[A_{n+1}\left(x\right)\beta (x;n)+B_{n+1}\left(x\right)\right]-A_n\left(x\right)A_{n+1}\left(x\right)\gamma (x;n)\right\}}{\mathfrak{D}(x;n)},$$

(28)

where

$$\mathfrak{D}(x;n)=B_{n-1}\left(x\right)\left[A_n\left(x\right)\beta (x;n-1)+B_n\left(x\right)\right]-A_{n-1}\left(x\right)A_n\left(x\right)\gamma (x;n-1).$$

(29)

As a consequence, we can state the following result.

Theorem 2.

Taking into account (29), the rational coefficients $\tilde{\beta }(x;n)$ and $\tilde{\gamma }(x;n)$ in (25) can be simplified to

$$\tilde{\beta }(x;n)=\frac{-B_{n-1}\left(x\right)}{A_n\left(x\right)}\frac{\mathfrak{D}(x;n+1)}{\mathfrak{D}(x;n)}+\frac{B_{n+1}\left(x\right)+A_{n+1}\left(x\right)\beta (x;n)}{A_n\left(x\right)},$$

(30)

and

$$\tilde{\gamma }(x;n)=\gamma (x;n-1)\frac{\mathfrak{D}(x;n+1)}{\mathfrak{D}(x;n)}.$$

(31)

Proof. 

Notice that (30) is

$$\tilde{\beta }(x;n)\mathfrak{D}(x;n)A_n\left(x\right)=$$

$$\begin{array}{ccc}& & \mathfrak{D}(x;n)B_{n+1}\left(x\right)+\mathfrak{D}(x;n)A_{n+1}\left(x\right)\beta (x;n)-B_{n-1}\left(x\right)\mathfrak{D}(x;n+1)\hfill \\ & =& A_{n+1}\left(x\right)A_n\left(x\right)B_{n-1}\left(x\right)\gamma (x;n)+A_{n+1}\left(x\right)A_n\left(x\right)B_{n-1}\left(x\right)\beta (x;n)\beta (x;n-1)\hfill \\ & & -A_{n+1}\left(x\right)A_n\left(x\right)A_{n-1}\left(x\right)\beta (x;n)\gamma (x;n-1)+A_n\left(x\right)B_{n+1}\left(x\right)B_{n-1}\left(x\right)\beta (x;n-1)\hfill \\ & & -A_n\left(x\right)A_{n-1}\left(x\right)B_{n+1}\left(x\right)\gamma (x;n-1)\hfill \\ & =& A_{n+1}\left(x\right)A_n\left(x\right)B_{n-1}\left(x\right)\gamma (x;n)\hfill \\ & & +A_n\left(x\right)\left[A_{n+1}\left(x\right)\beta (x;n)+B_{n+1}\left(x\right)\right]·\left[B_{n-1}\left(x\right)\beta (x;n-1)-A_{n-1}\left(x\right)\gamma (x;n-1)\right].\hfill \end{array}$$

The above expression may be rewritten exactly as (27), so this completes the proof of (30).

Next, from (28) we get

$$\frac{\mathfrak{D}(x;n)\tilde{\gamma }(x;n)}{\gamma (x;n-1)}=B_n\left(x\right)\left[A_{n+1}\left(x\right)\beta (x;n)+B_{n+1}\left(x\right)\right]-A_n\left(x\right)A_{n+1}\left(x\right)\gamma (x;n),$$

and notice that the right hand side of the above equation is just $\mathfrak{D}(x;n+1)$, so (31) is also proved. □

3. Asymptotics for the Recurrence Coefficients: The Freud Case

The study of the sequence of polynomials orthogonal with respect to the inner product

$$⟨p,q⟩={\int }_{\mathbb{R}}p\left(x\right)q\left(x\right)e^{-x^4}dx,p,q\in \mathbb{P},$$

(32)

was initiated by P. Nevai in [15,16]. In this case, the associated polynomials ${\left\{F_n\left(x\right)\right\}}_{n\ge 0}$ are called Freud polynomials, and they belong to the class of semiclassical orthogonal polynomials (see [17,18]). Freud orthogonal polynomials have been widely studied in the literature, mainly in relation to the coefficients in their corresponding three-term recurrence relation. Dealing with (continuous) orthogonal polynomials on the real line, one has explicit and simple expressions of these recurrence coefficients only for the case of the so-called classical orthogonal polynomials (i.e., Jacobi, Laguerre, and Hermite families). The next simplest situation where one can compute the coefficients of the recurrence relation, without having explicit expressions for them, occurs when Freud families (symmetric with respect to the origin) are considered, since their corresponding recurrence relation (when one considers the monic normalization) has only one coefficient $a_n$ (see (5)). In these families, $a_n$ is easily found as a solution to a very remarkable difference equation, usually known in the literature as “string equation”, or “Freud-like equation”. Within these Freud-like polynomials, the simplest case is precisely when the weight function is $e^{-x^4}$, whose string equation is formula (35) below. Furthermore, recent research has shown that these equations and their solutions (the recurrence coefficients $a_n$) are directly related to the well known six Painlevé trascendents ($P_I$ to $P_{VI})$, which constitutes at present a very active research line (see the nice recent contribution [19] and references therein).

In this section, we consider the particular case of the so-called Freud–Sobolev polynomials, orthogonal with respect to

$${⟨p,q⟩}_s={\int }_{\mathbb{R}}p\left(x\right)q\left(x\right)e^{-x^4}d\left(x\right)+M_{0}p\left(0\right)q\left(0\right)+M_1{p}^{\prime }\left(0\right)q^{\prime }\left(0\right),$$

(33)

i.e., the inner product (2) with $d\mu \left(x\right)=e^{-x^4}dx$. We point out that this Sobolev-type inner product has been studied previously in the arXiv preprint [10,11]. In the former, the author obtains connection formulas, a rescaled relative asymptotic (a kind of Plancherel–Rotach asymptotic result, see for example [20]), and interlacing results for zeros of ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$, all of them when $M_1=0$ (in Section 2 in [10]) and also when $M_1>0$ (in Section 3 in [10]). In addition, in Proposition 3.5 in [10] the existence of a five term recurrence relation that polynomials in ${\left\{Q_n\left(x\right)\right\}}_{n\ge 0}$ satisfy is stated, but the author does not provide any expression at all for the recurrence coefficients. [11], on the other hand, is focused in algebraic properties such as connection formulas and some computational aspects for their zeros (based on their role as eigenvalues of the corresponding Hessenberg matrix) as well as their behavior for large values of $M_0$ and $M_1$. Here, we analyze the asymptotic behavior of some of the coefficients of the recurrence relations considered in the previous section. In this sense, this particular section can be seen as a continuation of [11].

Asymptotic properties for these particular polynomials are well-known in the literature. In order to deduce our asymptotic results, we state some of them in the following proposition.

Proposition 2.

If $d\mu \left(x\right)=e^{-x^4}dx$, then the coefficients on the recurrence relation (5) satisfy

$\left(i\right)$

Asymptotics of $a_n^2$ (see [16,21])

$$a_n^2={\left(\frac{n}{12}\right)}^{\frac{1}{2}}\left[1+\frac{1}{24n^2}+\mathcal{O}\left(n^{-4}\right)\right].$$

(34)

$\left(ii\right)$

String equation (see (2.12) in [22]). ${\left\{a_n\right\}}_{n\ge 1}$ satisfies the following nonlinear difference equation

$$4a_n^2\left(a_{n+1}^2+a_n^2+a_{n-1}^2\right)=n,n\ge 1.$$

(35)

This is known in the literature as the string equation or Freud equation (see [23] and (3.2.20) in [24], among others).

$\left(iii\right)$

Asymptotics for $f_n\left(0\right)$ (see Th. 6 in [15]). There exists a constant $A=\sqrt[8]{12}/\sqrt{\pi }$ such that the following estimates hold

$$\begin{array}{ccc}\hfill f_n\left(0\right)& =& \left\{\begin{array}{cc}0\hfill & ifnisodd\hfill \\ {(-1)}^{n/2}{n}^{-1/8}(A+o\left(1\right))\hfill & ifniseven\hfill \end{array}\right.,\hfill \\ \hfill {}_n{]}^{\prime }\left(0\right)& =& \left\{\begin{array}{cc}{(-1)}^{(n-1)/2}\frac{\sqrt{8}}{\sqrt[4]{27}}{n}^{5/8}(A+o\left(1\right))\hfill & ifnisodd\hfill \\ 0\hfill & ifniseven\hfill \end{array}\right.,\hfill \\ \hfill {}_n{]}^{\prime \prime }\left(0\right)& =& \left\{\begin{array}{cc}0\hfill & ifnisodd\hfill \\ {\left(-1\right)}^{\frac{n}{2}+1}\frac{8}{3\sqrt{3}}{n}^{11/8}(A+o\left(1\right))\hfill & ifniseven\hfill \end{array}\right.,\hfill \\ \hfill {}_n{]}^{\prime \prime \prime }\left(0\right)& =& \left\{\begin{array}{cc}{(-1)}^{(n-1)/2}\frac{16\sqrt{2}}{9\sqrt[4]{3}}{n}^{17/8}(A+o\left(1\right))\hfill & ifnisodd\hfill \\ 0\hfill & ifniseven\hfill \end{array}\right..\hfill \end{array}$$

We also need explicit asymptotic expressions for the reproducing kernel and their derivatives. They are introduced in the following lemma.

Lemma 2.

For every $n=0,1,\dots$, we have

$$\begin{array}{ccc}\hfill K_n(0,0)& =& \mathcal{O}\left(n^{3/4}\right),\hfill \\ \hfill K_n^{(1,1)}(0,0)& =& \mathcal{O}\left(n^{9/4}\right).\hfill \end{array}$$

Proof. 

It suffices to write $K_n(0,0)$ and $K_n^{(1,1)}(0,0)$ in terms of the orthonormal polynomials $f_n$ and use the asymptotic results in $\left(iii\right)$ of Proposition 2. □

3.1. Asymptotics of the Recurrence Coefficients

We now proceed to analyze the asymptotic behavior of the coefficients in the five-term recurrence relation. First, we need the following lemma.

Lemma 3.

We have

$$\underset{n\to \infty }{lim}\frac{\parallel Q_n{\parallel }_s}{\parallel F_n\parallel }=1+\mathcal{O}\left(n^{-1}\right).$$

(36)

Proof. 

Let us consider first the even case. From (3) and its analogue for $Q_n$, as well as Proposition 1, we obtain

$$\frac{\parallel Q_{2n}{\parallel }_s^2}{\parallel F_{2n}{\parallel }^2}=\frac{1+M_0{K}_{2n}(0,0)}{1+M_0{K}_{2n-2}(0,0)}=\frac{1+M_0(K_{2n-2}(0,0)+f_{2n}^2\left(0\right))}{1+M_0{K}_{2n-2}(0,0)}.$$

Taking into account $\left(iii\right)$ of Proposition 2 and Lemma 2, the result follows. The odd case is similar. □

Notice that successive applications of the three-term recurrence relation (5) yield

$$x^2{F}_n\left(x\right)=F_{n+2}\left(x\right)+[a_{n+1}^2+a_n^2]F_n\left(x\right)+a_{n-1}^2{a}_n^2{F}_{n-2}\left(x\right).$$

We will show that, when $n\to \infty$, the five term recurrence relation (18) behaves exactly as the previous equation.

Proposition 3.

We have

$$\underset{n\to \infty }{lim}\frac{{\lambda }_{n,n}}{a_{n+1}^2+a_n^2}=1+\mathcal{O}\left(n^{-2}\right),and\underset{n\to \infty }{lim}\frac{{\lambda }_{n,n-2}}{a_{n-1}^2{a}_n^2}=1+\mathcal{O}\left(n^{-3/2}\right).$$

Proof. 

In view of (18) and (36), we need estimates for ${lim}_{n\to \infty }{\mathcal{A}}_{10}\left(n\right)$ and ${lim}_{n\to \infty }{\mathcal{B}}_{11}\left(n\right)$. It is easy to show that ${\mathcal{A}}_{10}\left(2n\right)=0$, and for the odd case we have

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{10}(2n+1)& =& -\frac{M_1{\left[Q_{2n+1}\right]}^{\prime }\left(0\right)F_{2n}\left(0\right)}{\left|\right|F_{2n}{\left|\right|}^2}=-\frac{M_1\left(\frac{{\left[F_{2n+1}\right]}^{\prime }\left(0\right)}{1+M_1{K}_{2n-1}^{(1,1)}(0,0)}\right)F_{2n}\left(0\right)}{\left|\right|F_{2n}{\left|\right|}^2}\hfill \\ & =& -\frac{M_1{K}_{2n}(0,0)}{1+M_1{K}_{2n-1}^{(1,1)}(0,0)},\hfill \end{array}$$

where the second equality follows from (11) and the third equality from the confluent expression (7). As a consequence, using Lemma 2, we get ${\mathcal{A}}_{10}(2n+1)=\mathcal{O}\left(n^{-3/2}\right)$. On the other side, for the even case,

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{11}\left(2n\right)& =& \frac{M_0{Q}_{2n}\left(0\right)F_{2n}\left(0\right)}{\left|\right|F_{2n-1}{\left|\right|}^2}=\frac{M_0{F}_{2n}^2\left(0\right)}{\left|\right|F_{2n-1}{\left|\right|}^2(1+M_0{K}_{2n-2}(0,0))}\hfill \\ & =& \frac{M_0{\parallel F_{2n}\parallel }^2{f}_{2n}^2\left(0\right)}{\parallel F_{2n-1}{\parallel }^2(1+M_0{K}_{2n-2}(0,0))},\hfill \end{array}$$

where we have used (11) for the second equality, and for the third equality the fact that $F_{2n}^2\left(0\right)=\parallel F_{2n}{\parallel }^2(K_{2n}(0,0)-K_{2n-1}(0,0))={\parallel F_{2n}\parallel }^2{f}_{2n}^2\left(0\right)$. By using (34), $\left(iii\right)$ of Proposition 2, Lemma 2 and $\parallel F_{2n}{\parallel }^2/{\parallel F_{2n-1}\parallel }^2=a_{2n}^2$, we obtain ${\mathcal{B}}_{11}\left(2n\right)=\mathcal{O}\left(n^{-1/2}\right)$. Finally, for the odd case, in a similar way we have

$${\mathcal{B}}_{11}(2n+1)=\frac{M_1{\left({\left[F_{2n+1}\right]}^{\prime }\left(0\right)\right)}^2}{\left|\right|F_{2n}{\left|\right|}^2(1+M_1{K}_{2n-1}^{(1,1)}(0,0))}=\frac{M_1{\parallel F_{2n+1}\parallel }^2{\left(f_{2n+1}^{\prime }\left(0\right)\right)}^2}{\parallel F_{2n}{\parallel }^2(1+M_1{K}_{2n-1}^{(1,1)}(0,0))},$$

and again from (34), $\left(iii\right)$ of Proposition 2 and Lemma 2, we get ${\mathcal{B}}_{11}(2n+1)=\mathcal{O}\left(n^{-1/2}\right)$.

As a consequence, we have

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\frac{{\lambda }_{n,n}}{a_{n+1}^2+a_n^2}& =& \underset{n\to \infty }{lim}\frac{\left[a_{n+1}^2+a_n^2+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right)\right]\frac{\left|\right|F_n{\left|\right|}^2}{\left|\right|Q_n{\left|\right|}_1^2}+{\mathcal{A}}_{10}\left(n\right)+{\mathcal{B}}_{11}\left(n\right),}{a_{n+1}^2+a_n^2}\hfill \\ & =& 1+\mathcal{O}\left(n^{-2}\right),\hfill \end{array}$$

and

$$\underset{n\to \infty }{lim}\frac{{\lambda }_{n,n-2}}{a_{n-1}^2{a}_n^2}=\frac{a_{n-1}^2\left[a_n^2+{\mathcal{B}}_{11}\left(n\right)\right]\frac{\left|\right|F_{n-2}{\left|\right|}^2}{\left|\right|Q_{n-2}{\left|\right|}_1^2}.}{a_{n-1}^2{a}_n^2}=1+\mathcal{O}\left(n^{-3/2}\right).$$

□

3.2. Asymptotics for the Rational Recurrence Coefficients

From (30) and (31), it is now possible to compute the asymptotic behavior when $n\to \infty$ of $\tilde{\beta }(x;n)$ and $\tilde{\gamma }(x;n)$, for the Freud–Sobolev type orthogonal polynomials $Q_n\left(x\right)$. For the Freud polynomials $F_n\left(x\right)$ we have $\beta (x;n)=x$ and $\gamma (x;n)=-a_n^2$, which combined with (13) yield

$$\mathfrak{D}(x;n)={\mathcal{B}}_{11}(n-1)\left(1+\frac{{\mathcal{A}}_{10}\left(n\right)}{x^2}+\frac{{\mathcal{B}}_{11}\left(n\right)}{x^2}\right)+\left(1+\frac{{\mathcal{A}}_{10}(n-1)}{x^2}\right)\left(1+\frac{{\mathcal{A}}_{10}\left(n\right)}{x^2}\right)a_{n-1}^2.$$

From the proof of Proposition 3 we also know

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}{\mathcal{B}}_{11}\left(n\right)& =& \mathcal{O}\left(n^{-1/2}\right),\hfill \\ \hfill \underset{n\to \infty }{lim}{\mathcal{A}}_{10}(2n+1)& =& \mathcal{O}\left(n^{-3/2}\right),{\mathcal{A}}_{10}\left(2n\right)=0,\hfill \end{array}$$

which together with (34) yield

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\mathfrak{D}(x;n)& =& \mathcal{O}\left(n^{1/2}\right),\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\frac{\mathfrak{D}(x;n+1)}{\mathfrak{D}(x;n)}& =& 1.\hfill \end{array}$$

(38)

Thus, we finally obtain

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}\tilde{\beta }(x;n)& =& \underset{n\to \infty }{lim}\left(\frac{-B_{n-1}\left(x\right)}{A_n\left(x\right)}\frac{\mathfrak{D}(x;n+1)}{\mathfrak{D}(x;n)}+\frac{B_{n+1}\left(x\right)+A_{n+1}\left(x\right)\beta (x;n)}{A_n\left(x\right)}\right)\hfill \\ & =& \underset{n\to \infty }{lim}\left(\frac{-\mathcal{O}\left(n^{-1/2}\right)}{x^2+\mathcal{O}\left(n^{-3/2}\right)}+\frac{x^3+\mathcal{O}\left(n^{-1/2}\right)+\mathcal{O}\left(n^{-3/2}\right)}{x^2+\mathcal{O}\left(n^{-3/2}\right)}\right)\hfill \\ & =& x,\hfill \\ \hfill \underset{n\to \infty }{lim}\tilde{\gamma }(x;n)& =& \underset{n\to \infty }{lim}\left(-a_{n-1}^2\frac{\mathfrak{D}(x;n+1)}{\mathfrak{D}(x;n)}\right)=\mathcal{O}\left(n^{\frac{1}{2}}\right).\hfill \end{array}$$

Thus, we have proved the following result.

Proposition 4.

For the Freud–Sobolev type orthogonal polynomials, the asymptotic behavior as n goes to infinity of the rational coefficients in their three-term recurrence formula (24) are

$$\underset{n\to \infty }{lim}\tilde{\beta }(x;n)=x,and\underset{n\to \infty }{lim}\tilde{\gamma }(x;n)=\mathcal{O}\left(n^{\frac{1}{2}}\right).$$

That is, these asymptotic behaviors coincide with the asymptotics of the coefficients $\beta (x;n)=x$, and $\gamma (x;n)=-a_n^2$ for Freud polynomials $F_n\left(x\right)$ in the corresponding three term recurrence formula with rational coefficients

$$F_{n+1}\left(x\right)=\beta (x;n)F_n\left(x\right)+\gamma (x;n)F_{n-1}\left(x\right)$$

which is just a simple rewriting of (5).

Finally, to complete this section and as a consequence of Theorem 2, we deduce a new type of difference equation which relates the values of the rational coefficients $\tilde{\gamma }(x;n)$ in (24), in the same way as the string Equation (35) relates the values of coefficients $a_n$ in (5).

Proposition 5 (Modified string equation).

For $n\ge 1$, we have

$$4\frac{\mathfrak{D}(x;n+1)}{\mathfrak{D}(x;n+2)}\tilde{\gamma }(x;n+1)\times$$

$$\left(\frac{\mathfrak{D}(x;n+2)}{\mathfrak{D}(x;n+3)}\tilde{\gamma }(x;n+2)+\frac{\mathfrak{D}(x;n+1)}{\mathfrak{D}(x;n+2)}\tilde{\gamma }(x;n+1)+\frac{\mathfrak{D}(x;n)}{\mathfrak{D}(x;n+1)}\tilde{\gamma }(x;n)\right)=n.$$

Proof. 

Combining (31) with $\gamma (x;n)=-a_n^2$ we deduce

$$a_{n-1}^2=\frac{-\mathfrak{D}(x;n)}{\mathfrak{D}(x;n+1)}\tilde{\gamma }(x;n).$$

Shifting the index n when necessary, and replacing the above in (35) immediately gives the result for $n\ge 1$. □

Notice that the above proposition allows one to obtain the coefficient $\tilde{\gamma }(x;n+2)$, knowing the values of the two precedent coefficients $\tilde{\gamma }(x;n+1)$ and $\tilde{\gamma }(x;n)$.

4. Conclusions and Further Discussion

In this contribution, we have obtained several connection formulas to deduce a five-term recurrence relation as well as a three-term recurrence relation with rational coefficients for Sobolev-type orthogonal polynomials (see Theorems 1 and 2 in Section 2). These recurrence relations are very general in the sense that they are valid for any symmetric weight $\mu$, as they depend only on the symmetry properties of the orthogonality weight. We also have used these relations to deduce the asymptotic behavior of its coefficients, in the particular case of the Freud–Sobolev orthogonal polynomials (Propositions 3 and 4). It was determined that, asymptotically, the (rational) coefficients of the three-term recurrence relation for the Freud–Sovolev polynomials behave exactly as the corresponding coefficients for the Freud polynomials. The same occurs for the coefficients of the five-recurrence relation. We have also derived a nonlinear difference equation for the rational coefficients of the recurrence relation for the Freud–Sobolev orthogonal polynomials. Notice that, by using similar techniques, the asymptotic results in Section 3 can be obtained for any symmetric weight function, provided the corresponding asymptotic properties (the analogue of Proposition 2 for the associated orthogonal polynomials) are known. An interesting open question is whether or not the asymptotic behavior of the recurrence coefficients for both the Sobolev-type and the unperturbed orthogonal polynomials will coincide for any symmetric orthogonality weight. This problem will be addressed in a future contribution.