The Recurrence Coefficients of Orthogonal Polynomials with a Weight Interpolating between the Laguerre Weight and the Exponential Cubic Weight

Abstract

In this paper, we consider the orthogonal polynomials with respect to the weight $w\left(x\right)=w(x;s):=x^{\lambda }{\mathrm{e}}^{-N[x+s(x^3-x)]},x\in {\mathbb{R}}^{+},$ where $\lambda >0$, $N>0$ and $0\le s\le 1$. By using the ladder operator approach, we obtain a pair of second-order nonlinear difference equations and a pair of differential–difference equations satisfied by the recurrence coefficients ${\alpha }_n\left(s\right)$ and ${\beta }_n\left(s\right)$. We also establish the relation between the associated Hankel determinant and the recurrence coefficients. From Dyson’s Coulomb fluid approach, we prove that the recurrence coefficients converge and the limits are derived explicitly when $q:=n/N$ is fixed as $n\to \infty$.

1. Introduction

In this paper, we are concerned with the coefficients in the three-term recurrence relation for the orthogonal polynomials with respect to the weight

$$w\left(x\right)=w(x;s):=x^{\lambda }{\mathrm{e}}^{-N[x+s(x^3-x)]},x\in {\mathbb{R}}^{+},$$

(1)

with parameters $\lambda >0$, $N>0$ and $0\le s\le 1$.

If $s=0$, the weight (1) is the classical (scaled with N) Laguerre weight. If $s=1$, it is an exponential cubic weight. Orthogonal polynomials associated with the exponential cubic weight have been well studied (see e.g., [1,2,3,4]), and have important applications in numerical analysis [5] and random matrix theory [6,7,8]. Furthermore, orthogonal polynomials and the Hankel determinant for the so-called semi-classical Laguerre weight $\tilde{w}\left(x\right)=x^{\lambda }{\mathrm{e}}^{-N[x+s(x^2-x)]},x\in {\mathbb{R}}^{+}$ have been studied in [9,10], which is also the motivation of the present paper.

Let ${\left\{P_n(x;s)\right\}}_{n=0}^{\infty }$ be a sequence of monic polynomials, $P_n\left(x\right)$ of degree n, orthogonal with respect to the weight (1); that is,

$${\int }_0^{\infty }{P}_m(x;s)P_n(x;s)w(x;s)dx=h_n\left(s\right){\delta }_{mn},m,n=0,1,2,\dots ,$$

(2)

where $h_n\left(s\right)>0$ and $P_n(x;s)$ has the expansion

$$P_n(x;s)=x^n+\mathbf{p}(n,s)x^{n-1}+\cdots +P_n(0;s),$$

where $\mathbf{p}(n,s)$, the sub-leading coefficient of $P_n(x;s)$, will play a significant role in the following discussions. Note that $P_n(x;s)$ and $\mathbf{p}(n,s)$ also depend on the parameters $\lambda$ and N.

One of the most important properties of the orthogonal polynomials is that they satisfy the three-term recurrence relation of the form

$$x{P}_n(x;s)=P_{n+1}(x;s)+{\alpha }_n\left(s\right)P_n(x;s)+{\beta }_n\left(s\right)P_{n-1}(x;s),$$

(3)

with the initial conditions

$$P_0(x;s):=1,{\beta }_0\left(s\right)P_{-1}(x;s):=0.$$

As an easy consequence, we have

$${\alpha }_n\left(s\right)=\mathbf{p}(n,s)-\mathbf{p}(n+1,s),$$

(4)

$${\beta }_n\left(s\right)={\displaystyle \frac{h_n\left(s\right)}{h_{n-1}\left(s\right)}}>0.$$

(5)

Taking a telescopic sum of (4) and noting that $\mathbf{p}(0,s):=0$, we obtain an important identity

$$\sum _{j=0}^{n-1}{\alpha }_j\left(s\right)=-\mathbf{p}(n,s).$$

(6)

It is known that (see, e.g., [11] (p. 17)) $P_n(x;s)$ can be expressed as the determinant

$$P_n(x;s)=\frac{1}{D_n\left(s\right)}\left|\begin{array}{cccc}{\mu }_0\left(s\right)& {\mu }_1\left(s\right)& \cdots & {\mu }_n\left(s\right)\\ {\mu }_1\left(s\right)& {\mu }_2\left(s\right)& \cdots & {\mu }_{n+1}\left(s\right)\\ ⋮& ⋮& & ⋮\\ {\mu }_{n-1}\left(s\right)& {\mu }_n\left(s\right)& \cdots & {\mu }_{2n-1}\left(s\right)\\ 1& x& \cdots & x^n\end{array}\right|$$

and

$$h_n\left(s\right)=\frac{D_{n+1}\left(s\right)}{D_n\left(s\right)},$$

(7)

where $D_n\left(s\right)$ is the Hankel determinant for the weight (1) defined by

$$D_n\left(s\right):=det{\left({\mu }_{i+j}\left(s\right)\right)}_{i,j=0}^{n-1}=\left|\begin{array}{cccc}{\mu }_0\left(s\right)& {\mu }_1\left(s\right)& \cdots & {\mu }_{n-1}\left(s\right)\\ {\mu }_1\left(s\right)& {\mu }_2\left(s\right)& \cdots & {\mu }_n\left(s\right)\\ ⋮& ⋮& & ⋮\\ {\mu }_{n-1}\left(s\right)& {\mu }_n\left(s\right)& \cdots & {\mu }_{2n-2}\left(s\right)\end{array}\right|,$$

and ${\mu }_j\left(s\right)$ is the jth moment given by the integral

$${\mu }_j\left(s\right):={\int }_0^{\infty }{x}^{j}w(x;s)dx.$$

We mention that the moment ${\mu }_j\left(s\right)$ can be expressed in terms of the generalized hypergeometric functions after some calculations.

Furthermore, it is easy to see from (7) that the Hankel determinant $D_n\left(s\right)$ can be expressed as the product of $h_j\left(s\right)$ in the form

$$D_n\left(s\right)=\prod _{j=0}^{n-1}{h}_j\left(s\right).$$

(8)

Obviously, the recurrence coefficients ${\alpha }_n\left(s\right),{\beta }_n\left(s\right)$ and the Hankel determinant $D_n\left(s\right)$ are all dependent on the parameters $\lambda$ and N in our problem. For more information about orthogonal polynomials, see [11,12,13].

The remainder of the paper is organized as follows. In Section 2, by using the ladder operator approach, we derive the discrete system for the recurrence coefficients ${\alpha }_n\left(s\right)$ and ${\beta }_n\left(s\right)$. We also obtain an important identity in the representation of the sub-leading coefficient $\mathbf{p}(n,s)$ in terms of the recurrence coefficients. In Section 3, we derive the differential–difference equations satisfied by the recurrence coefficients. We establish the relation between the Hankel determinant $D_n\left(s\right)$ and the recurrence coefficients, and also obtain the differential–difference equations satisfied by $D_n\left(s\right)$. In Section 4, by making use of Dyson’s Coulomb fluid approach, we find that the large n limits of the recurrence coefficients exist in the sense that $n/N$ is fixed as $n\to \infty$. The expressions of the limits are also given explicitly. Finally, the conclusions and some remarks are outlined in Section 5.

2. Ladder Operators and Second-Order Difference Equations

The ladder operator approach has been applied to solve a series of problems about semi-classical orthogonal polynomials and the related Hankel determinants, especially the relationship to Painlevé equations; see, e.g., [14,15,16] and the references therein. Note that, in order to simplify the notations, the s-dependence of many quantities such as $P_n\left(x\right)$, $w\left(x\right)$, $h_n$, ${\alpha }_n$ and ${\beta }_n$ will not be displayed unless it is needed. Following the general set-up of Chen and Ismail [17,18], the lowering and raising operators for our orthogonal polynomials are

$$\left({\displaystyle \frac{d}{dx}}+B_n\left(x\right)\right)P_n\left(x\right)={\beta }_n{A}_n\left(x\right)P_{n-1}\left(x\right),$$

$$\left({\displaystyle \frac{d}{dx}}-B_n\left(x\right)-{\mathrm{v}}^{\prime }\left(x\right)\right)P_{n-1}\left(x\right)=-A_{n-1}\left(x\right)P_n\left(x\right),$$

where the functions $A_n\left(x\right)$ and $B_n\left(x\right)$ are defined by

$$A_n\left(x\right):={\displaystyle \frac{1}{h_n}}{\int }_0^{\infty }{\displaystyle \frac{{\mathrm{v}}^{\prime }\left(x\right)-{\mathrm{v}}^{\prime }\left(y\right)}{x-y}}{P}_n^2\left(y\right)w\left(y\right)dy,$$

(9)

$$B_n\left(x\right):={\displaystyle \frac{1}{h_{n-1}}}{\int }_0^{\infty }{\displaystyle \frac{{\mathrm{v}}^{\prime }\left(x\right)-{\mathrm{v}}^{\prime }\left(y\right)}{x-y}}{P}_n\left(y\right)P_{n-1}\left(y\right)w\left(y\right)dy,$$

(10)

and $\mathrm{v}\left(x\right)=-lnw\left(x\right)$.

The associated compatibility conditions for the functions $A_n\left(x\right)$ and $B_n\left(x\right)$ are

$$B_{n+1}\left(x\right)+B_n\left(x\right)=(x-{\alpha }_n)A_n\left(x\right)-{\mathrm{v}}^{\prime }\left(x\right),$$

(11)

$$1+(x-{\alpha }_n)(B_{n+1}\left(x\right)-B_n\left(x\right))={\beta }_{n+1}{A}_{n+1}\left(x\right)-{\beta }_n{A}_{n-1}\left(x\right),$$

(12)

$$B_n^2\left(x\right)+{\mathrm{v}}^{\prime }\left(x\right)B_n\left(x\right)+\sum _{j=0}^{n-1}{A}_j\left(x\right)={\beta }_n{A}_n\left(x\right)A_{n-1}\left(x\right).$$

(13)

Here, (13) is obtained by the combination of (11) and (12), and is usually more useful compared to (12).

For our problem with the weight (1), we have

$$\mathrm{v}\left(x\right)=-lnw\left(x\right)=N[x+s(x^3-x)]-\lambda lnx,$$

and

$${\displaystyle \frac{{\mathrm{v}}^{\prime }\left(x\right)-{\mathrm{v}}^{\prime }\left(y\right)}{x-y}}=3Ns(x+y)+{\displaystyle \frac{\lambda }{xy}}.$$

(14)

Using (14), we compute the functions $A_n\left(x\right)$ and $B_n\left(x\right)$ in the following lemma.

Lemma 1.

For our problem, the expressions of $A_n\left(x\right)$ and $B_n\left(x\right)$ are given by

$$A_n\left(x\right)=3Ns(x+{\alpha }_n)+{\displaystyle \frac{R_n\left(s\right)}{x}},$$

(15)

$$B_n\left(x\right)=3Ns{\beta }_n+{\displaystyle \frac{r_n\left(s\right)}{x}},$$

(16)

where $R_n\left(s\right)$ and $r_n\left(s\right)$ are the auxiliary quantities

$$R_n\left(s\right):={\displaystyle \frac{\lambda }{h_n}}{\int }_0^{\infty }{\displaystyle \frac{1}{y}}{P}_n^2\left(y\right)w\left(y\right)dy,$$

$$r_n\left(s\right):={\displaystyle \frac{\lambda }{h_{n-1}}}{\int }_0^{\infty }{\displaystyle \frac{1}{y}}{P}_n\left(y\right)P_{n-1}\left(y\right)w\left(y\right)dy.$$

Proof. 

Substituting (14) into the definitions of $A_n\left(x\right)$ and $B_n\left(x\right)$ in (9) and (10), we obtain the desired results by using the orthogonality condition (2) and the three-term recurrence relation (3). □

From the compatibility conditions (11) and (13), we have the following results.

Proposition 1.

The recurrence coefficients ${\alpha }_n,{\beta }_n$ and the auxiliary quantities $R_n\left(s\right),r_n\left(s\right)$ satisfy the relations as follows:

$$3Ns({\beta }_{n+1}+{\beta }_n)=R_n\left(s\right)-N(1-s)-3Ns{\alpha }_n^2,$$

(17)

$$r_{n+1}\left(s\right)+r_n\left(s\right)=\lambda -{\alpha }_n{R}_n\left(s\right),$$

(18)

$$r_n\left(s\right)+n=3Ns{\beta }_n({\alpha }_n+{\alpha }_{n-1}),$$

(19)

$$3Ns{\beta }_n^2+N(1-s){\beta }_n+\sum _{j=0}^{n-1}{\alpha }_j={\beta }_n\left(R_n\left(s\right)+R_{n-1}\left(s\right)+3Ns{\alpha }_n{\alpha }_{n-1}\right),$$

(20)

$$N{r}_n\left(s\right)(6s{\beta }_n+1-s)+\sum _{j=0}^{n-1}{R}_j\left(s\right)=3Ns{\beta }_n\left({\alpha }_n{R}_{n-1}\left(s\right)+{\alpha }_{n-1}{R}_n\left(s\right)+\lambda \right),$$

(21)

$$r_n^2\left(s\right)-\lambda r_n\left(s\right)={\beta }_n{R}_n\left(s\right)R_{n-1}\left(s\right).$$

(22)

Proof. 

Substituting (15) and (16) into (11), and comparing the coefficients of $z^0$ and $z^{-1}$ on both sides, we obtain (17) and (18), respectively. Similarly, substituting (15) and (16) into (13), and comparing the coefficients of $z^1,z^0,z^{-1}$ and $z^{-2}$ on both sides, we obtain (19), (20), (21) and (22), respectively. □

Now we are ready to derive the main result of this section on the discrete system for the recurrence coefficients.

Theorem 1.

The recurrence coefficients ${\alpha }_n$ and ${\beta }_n$ satisfy a pair of second-order nonlinear difference equations:

$$3s\left[{\alpha }_n^3+{\beta }_n(2{\alpha }_n+{\alpha }_{n-1})+{\beta }_{n+1}(2{\alpha }_n+{\alpha }_{n+1})\right]+(1-s){\alpha }_n={\displaystyle \frac{2n+\lambda +1}{N}},$$

(23a)

$$\begin{array}{cc}& \left[3s{\beta }_n({\alpha }_n+{\alpha }_{n-1})-{\displaystyle \frac{n}{N}}\right]\left[3s{\beta }_n({\alpha }_n+{\alpha }_{n-1})-{\displaystyle \frac{n+\lambda }{N}}\right]={\beta }_n\left[3s({\alpha }_n^2+{\beta }_n+{\beta }_{n+1})+1-s\right]\hfill \\ & \times \left[3s({\alpha }_{n-1}^2+{\beta }_{n-1}+{\beta }_n)+1-s\right].\hfill \end{array}$$

(23b)

Proof. 

From (17) and (19), we can express $R_n\left(s\right)$ and $r_n\left(s\right)$ in terms of the recurrence coefficients:

$$R_n\left(s\right)=3Ns({\alpha }_n^2+{\beta }_n+{\beta }_{n+1})+N(1-s),$$

(24)

$$r_n\left(s\right)=3Ns{\beta }_n({\alpha }_n+{\alpha }_{n-1})-n.$$

(25)

Substituting (24) and (25) into (18) and (22), we obtain (23a) and (23b), respectively. □

Remark 1.

When $s=0$, the results in the above theorem are reduced to

$$N{\alpha }_n\left(0\right)=2n+\lambda +1,N^2{\beta }_n\left(0\right)=n(n+\lambda ),$$

(26)

which are consistent with the recurrence coefficients of the classical monic Laguerre polynomials.

At the end of this section, we give an expression of the sub-leading coefficient $\mathbf{p}(n,s)$, which will be very useful in the analysis of the next section.

Corollary 1.

The sub-leading coefficient $\mathbf{p}(n,s)$ can be expressed in terms of the recurrence coefficients as follows:

$$\mathbf{p}(n,s)=-N{\beta }_n\left[3s\left({\alpha }_{n-1}^2+{\alpha }_{n-1}{\alpha }_n+{\alpha }_n^2+{\beta }_{n-1}+{\beta }_n+{\beta }_{n+1}\right)+1-s\right].$$

(27)

Proof. 

Substituting (6) into (20), we have

$$\mathbf{p}(n,s)=3Ns{\beta }_n^2+N(1-s){\beta }_n-{\beta }_n\left(R_n\left(s\right)+R_{n-1}\left(s\right)+3Ns{\alpha }_n{\alpha }_{n-1}\right).$$

Eliminating $R_n\left(s\right)$ and $R_{n-1}\left(s\right)$ by (24), we obtain (27). □

3. S Evolution and Differential-Difference Equations

Note that all the quantities discussed in this paper, such as the recurrence coefficients ${\alpha }_n$ and ${\beta }_n$, depend on the parameter s. We consider the s evolution in this section.

We start from taking a derivative with respect to s in the equation

$$h_n\left(s\right)={\int }_0^{\infty }{P}_n^2(x;s)x^{\lambda }{\mathrm{e}}^{-N[x+s(x^3-x)]}dx,$$

which gives

$$\begin{array}{ccc}\hfill 3s\frac{d}{ds}ln{h}_n\left(s\right)& =& \frac{3Ns}{h_n}{\int }_0^{\infty }(x-x^3)P_n^2\left(x\right)w\left(x\right)dx\hfill \\ & =& \frac{3Ns}{h_n}{\int }_0^{\infty }x{P}_n^2\left(x\right)w\left(x\right)dx-\frac{3Ns}{h_n}{\int }_0^{\infty }{x}^3{P}_n^2\left(x\right)w\left(x\right)dx.\hfill \end{array}$$

(28)

By the three-term recurrence relation (3), we obtain the first term

$$\frac{3Ns}{h_n}{\int }_0^{\infty }x{P}_n^2\left(x\right)w\left(x\right)dx=3Ns{\alpha }_n$$

(29)

and the second term

$$\begin{array}{ccc}\hfill \frac{3Ns}{h_n}{\int }_0^{\infty }{x}^3{P}_n^2\left(x\right)w\left(x\right)dx& =& 3Ns\left[{\alpha }_n^3+{\beta }_n(2{\alpha }_n+{\alpha }_{n-1})+{\beta }_{n+1}(2{\alpha }_n+{\alpha }_{n+1})\right]\hfill \\ & =& 2n+\lambda +1-N(1-s){\alpha }_n,\hfill \end{array}$$

(30)

where we have used (23a) in the second step to simplify the result.

From (28), (29) and (30), it follows that

$$3s{\displaystyle \frac{d}{ds}}ln{h}_n\left(s\right)=N(1+2s){\alpha }_n-\left(2n+\lambda +1\right).$$

(31)

Using (5), we have

$$3s{\displaystyle \frac{d}{ds}}ln{\beta }_n\left(s\right)=3s{\displaystyle \frac{d}{ds}}ln{h}_n\left(s\right)-3s{\displaystyle \frac{d}{ds}}ln{h}_{n-1}\left(s\right)=N(1+2s)({\alpha }_n-{\alpha }_{n-1})-2;$$

that is,

$$3s{\beta }_n^{\prime }\left(s\right)={\beta }_n\left[N(1+2s)({\alpha }_n-{\alpha }_{n-1})-2\right].$$

On the other hand, differentiating with respect to s in the equation

$${\int }_0^{\infty }{P}_n(x;s)P_{n-1}(x;s)x^{\lambda }{\mathrm{e}}^{-N[x+s(x^3-x)]}dx=0$$

produces

$$3s{\displaystyle \frac{d}{ds}}\mathbf{p}(n,s)=\frac{3Ns}{h_{n-1}}{\int }_0^{\infty }{x}^3{P}_n\left(x\right)P_{n-1}\left(x\right)w\left(x\right)dx-\frac{3Ns}{h_{n-1}}{\int }_0^{\infty }x{P}_n\left(x\right)P_{n-1}\left(x\right)w\left(x\right)dx.$$

(32)

The first term is

$$\begin{array}{ccc}\hfill \frac{3Ns}{h_{n-1}}{\int }_0^{\infty }{x}^3{P}_n\left(x\right)P_{n-1}\left(x\right)w\left(x\right)dx& =& 3Ns{\beta }_n\left({\alpha }_n^2+{\alpha }_n{\alpha }_{n-1}+{\alpha }_{n-1}^2+{\beta }_{n+1}+{\beta }_n+{\beta }_{n-1}\right)\hfill \\ & =& -\mathbf{p}(n,s)-N(1-s){\beta }_n,\hfill \end{array}$$

(33)

where we have used (27) to simplify the result in the second equality. The second term reads

$$\frac{3Ns}{h_{n-1}}{\int }_0^{\infty }x{P}_n\left(x\right)P_{n-1}\left(x\right)w\left(x\right)dx=3Ns{\beta }_n.$$

(34)

Substituting (33) and (34) into (32), we find

$$3s{\displaystyle \frac{d}{ds}}\mathbf{p}(n,s)=-\mathbf{p}(n,s)-N(1+2s){\beta }_n.$$

(35)

Taking account of (4), it follows that

$$3s{\alpha }_n^{\prime }\left(s\right)=-{\alpha }_n+N(1+2s)({\beta }_{n+1}-{\beta }_n).$$

To sum up, we have the following theorem.

Theorem 2.

The recurrence coefficients ${\alpha }_n$ and ${\beta }_n$ satisfy the coupled differential–difference equations:

$$3s{\alpha }_n^{\prime }\left(s\right)=-{\alpha }_n+N(1+2s)({\beta }_{n+1}-{\beta }_n),$$

$$3s{\beta }_n^{\prime }\left(s\right)={\beta }_n\left[N(1+2s)({\alpha }_n-{\alpha }_{n-1})-2\right].$$

We also derive some results about the Hankel determinant $D_n\left(s\right)$ as follows.

Theorem 3.

The logarithmic derivative of the Hankel determinant is expressed in terms of the recurrence coefficients as follows:

$$3s\frac{d}{ds}ln{D}_n\left(s\right)=N^2(1+2s){\beta }_n\left[3s\left({\alpha }_{n-1}^2+{\alpha }_{n-1}{\alpha }_n+{\alpha }_n^2+{\beta }_{n-1}+{\beta }_n+{\beta }_{n+1}\right)+1-s\right]-n(n+\lambda ).$$

Proof. 

From (8) and (31), we have

$$\begin{array}{ccc}\hfill 3s\frac{d}{ds}ln{D}_n\left(s\right)& =& \sum _{j=0}^{n-1}3s\frac{d}{ds}ln{h}_j\left(s\right)\hfill \\ & =& \sum _{j=0}^{n-1}\left[N(1+2s){\alpha }_j-\left(2j+\lambda +1\right)\right].\hfill \end{array}$$

Taking account of (6) and using (27), we find

$$\begin{array}{ccc}\hfill 3s\frac{d}{ds}ln{D}_n\left(s\right)& =& -N(1+2s)\mathbf{p}(n,s)-n(n+\lambda )\hfill \\ & =& N^2(1+2s){\beta }_n\left[3s\left({\alpha }_{n-1}^2+{\alpha }_{n-1}{\alpha }_n+{\alpha }_n^2+{\beta }_{n-1}+{\beta }_n+{\beta }_{n+1}\right)+1-s\right]\hfill \\ & & -n(n+\lambda ).\hfill \end{array}$$

(36)

The proof is complete. □

Corollary 2.

The Hankel determinant $D_n\left(s\right)$ satisfies the differential–difference equation

$$9s^2(1+2s)\frac{d^2}{d{s}^2}ln{D}_n\left(s\right)+6s(2+s)\frac{d}{ds}ln{D}_n\left(s\right)+n(n+\lambda )(1-4s)=N^2{(1+2s)}^3\frac{D_{n+1}\left(s\right)D_{n-1}\left(s\right)}{D_n^2\left(s\right)}.$$

Proof. 

From (36), we have

$$\mathbf{p}(n,s)=-\frac{3s\frac{d}{ds}ln{D}_n\left(s\right)+n(n+\lambda )}{N(1+2s)}.$$

(37)

A combination of (5) and (7) gives

$${\beta }_n\left(s\right)=\frac{D_{n+1}\left(s\right)D_{n-1}\left(s\right)}{D_n^2\left(s\right)}.$$

(38)

Substituting (37) and (38) into (35), we obtain the desired result. □

4. Asymptotics of the Recurrence Coefficients

Recall that, for our problem, the weight function is

$$w\left(x\right)=x^{\lambda }{\mathrm{e}}^{-N[x+s(x^3-x)]},x\in {\mathbb{R}}^{+}$$

(39)

and the potential is

$$\mathrm{v}\left(x\right)=N[x+s(x^3-x)]-\lambda lnx,x\in {\mathbb{R}}^{+},$$

(40)

where $\lambda >0$, $N>0$ and $0\le s\le 1$.

In random matrix theory [19,20,21], it is known that our Hankel determinant $D_n\left(s\right)$ is equal to the partition function for the unitary random matrix ensemble associated with the weight (39) [11] (Corollary 2.1.3), i.e.,

$$D_n\left(s\right)=\frac{1}{n!}{\int }_{{(0,\infty )}^n}\prod _{1\le i<j\le n}{(x_i-x_j)}^2\prod _{k=1}^n{x}_k^{\lambda }{\mathrm{e}}^{-N[x_k+s(x_k^3-x_k)]}d{x}_k,$$

where ${\left\{x_j\right\}}_{j=1}^n$ are the eigenvalues of $n\times n$ Hermitian matrices from the ensemble with the joint probability density function

$$p(x_1,x_2,\dots ,x_n)=\frac{1}{n!D_n\left(s\right)}\prod _{1\le i<j\le n}{(x_i-x_j)}^2\prod _{k=1}^n{x}_k^{\lambda }{\mathrm{e}}^{-N[x_k+s(x_k^3-x_k)]}.$$

If we interpret ${\left\{x_j\right\}}_{j=1}^n$ as the positions of n charged particles, then the collection of particles can be approximated as a continuous fluid with an equilibrium density $\sigma \left(x\right)$ in the limit of large n according to Dyson’s Coulomb fluid approach [22]. Since our potential $\mathrm{v}\left(x\right)$ in (40) is convex for $x\in {\mathbb{R}}^{+}$, the density $\sigma \left(x\right)$ is supported on an single interval denoted by $(0,b)$; see Chen and Ismail [23] and also [24] (p. 198).

Following [23], the equilibrium density $\sigma \left(x\right)$ is obtained by minimizing the free energy functional

$$F\left[\sigma \right]:={\int }_0^b\sigma \left(x\right)\mathrm{v}\left(x\right)dx-{\int }_0^b{\int }_0^b\sigma \left(x\right)ln|x-y|\sigma \left(y\right)dxdy$$

subject to the normalization condition

$${\int }_0^b\sigma \left(x\right)dx=n.$$

(41)

Upon minimization, the density $\sigma \left(x\right)$ satisfies the integral equation

$$\mathrm{v}\left(x\right)-2{\int }_0^{b}ln|x-y|\sigma \left(y\right)dy=A,x\in (0,b),$$

where A is the Lagrange multiplier for the constraint (41). Taking a derivative with respect to x in the above equation gives the singular integral equation

$${\mathrm{v}}^{\prime }\left(x\right)-2P{\int }_0^b\frac{\sigma \left(y\right)}{x-y}dy=0,x\in (0,b),$$

(42)

where P denotes the Cauchy principal value. From the theory of singular integral equations [25], the solution of (42) is given by

$$\sigma \left(x\right)=\frac{1}{2{\pi }^2}\sqrt{\frac{b-x}{x}}P{\int }_0^b\frac{{\mathrm{v}}^{\prime }\left(y\right)}{y-x}\sqrt{\frac{y}{b-y}}dy.$$

(43)

Substituting (40) into (43) and after some elaborate computations, we find

$$\sigma \left(x\right)=\frac{N}{2\pi }\sqrt{\frac{b-x}{x}}\left[1+s\left(3x^2+\frac{3bx}{2}+\frac{9b^2}{8}-1\right)\right].$$

The normalization condition (41) then becomes

$$\frac{1}{32}Nb\left[15s{b}^2+8(1-s)\right]=n.$$

(44)

Motivated by the works [9,10], we consider the case that $q:=n/N$ is fixed when $n\to \infty$. Equation (44) is actually a cubic equation for b,

$$15s{b}^3+8(1-s)b-32q=0,$$

which has a unique real solution given by

$$b=\frac{2^{4/3}}{3\times 5^{2/3}s}\left[{\xi }^{1/3}-{10}^{1/3}s(1-s){\xi }^{-1/3}\right],$$

where

$$\xi =45q{s}^2+s\sqrt{5s\left[2+3(135q^2-2)s+6s^2-2s^3\right]}.$$

It was shown in Chen and Ismail [23] that, as $n\to \infty$,

$${\alpha }_n\left(s\right)={\displaystyle \frac{b}{2}}+O\left({\displaystyle \frac{{\partial }^{2}A}{\partial s\partial n}}\right),$$

$${\beta }_n\left(s\right)={\displaystyle \frac{b^2}{16}}\left(1+O\left({\displaystyle \frac{{\partial }^{3}A}{\partial n^3}}\right)\right).$$

Hence, we have the following theorem.

Theorem 4.

Let $q:=n/N$ be fixed when $n\to \infty$. Then, the limits of ${\alpha }_n$ and ${\beta }_n$ as $n\to \infty$ exist and are given by

$$\underset{n\to \infty }{lim}{\alpha }_n=\frac{2^{1/3}}{3\times 5^{2/3}s}\left[{\xi }^{1/3}-{10}^{1/3}s(1-s){\xi }^{-1/3}\right],$$

(45)

$$\underset{n\to \infty }{lim}{\beta }_n=\frac{2^{2/3}}{180\times 5^{1/3}{s}^2}\left[{\xi }^{2/3}+{10}^{2/3}{s}^2{(1-s)}^2{\xi }^{-2/3}-2\times {10}^{1/3}s(1-s)\right],$$

(46)

where

$$\xi =45q{s}^2+s\sqrt{5s\left[2+3(135q^2-2)s+6s^2-2s^3\right]}.$$

Remark 2.

It is an interesting phenomenon that the limits of the recurrence coefficients in (45) and (46) are independent of the parameter λ.

Remark 3.

When $s\to 0^{+}$, we find from (45) and (46) that

$$\underset{n\to \infty }{lim}{\alpha }_n=2q,\underset{n\to \infty }{lim}{\beta }_n=q^2,$$

which coincides with the classical results for the Laguerre polynomials; see (26).

Remark 4.

We conjecture that ${\alpha }_n$ and ${\beta }_n$ have the following large n asymptotic expansion

$${\alpha }_n=\sum _{j=0}^{\infty }\frac{a_j}{n^j},{\beta }_n=\sum _{j=0}^{\infty }\frac{b_j}{n^j},$$

where $a_0$ and $b_0$ are given by the right hand sides of (45) and (46), respectively. Then, one can determine the expansion coefficients $a_j$ and $b_j$ recursively by using the discrete system for the recurrence coefficients in (23) following the procedure in [14,15,16]. However, the results are too complicated to write down here.

5. Conclusions

In this paper, we studied the monic polynomials orthogonal with respect to a semi-classical weight, which interpolates between the classical Laguerre weight and the exponential cubic weight. By making use of the ladder operator approach, we derived the discrete system for the recurrence coefficients ${\alpha }_n\left(s\right)$ and ${\beta }_n\left(s\right)$. Considering the s evolution, we obtained the coupled differential–difference equations satisfied by ${\alpha }_n\left(s\right)$ and ${\beta }_n\left(s\right)$. We also studied the relations between the associated Hankel determinant, the sub-leading coefficient of the monic orthogonal polynomials and the recurrence coefficients. Finally, we proved that the large n limits of the recurrence coefficients exist and are given when $n/N$ is fixed as $n\to \infty$. The large n asymptotic expansions of the recurrence coefficients, the sub-leading coefficient $\mathbf{p}(n,s)$ and the Hankel determinant $D_n\left(s\right)$ in the sense that $n/N$ is fixed as $n\to \infty$ can be considered based on the results in this paper; however, we found that the computations are very cumbersome.