Orthogonality Property of the Discrete q-Hermite Matrix Polynomials

Abstract

In this paper, we prove that the solution of the autonomous $q$-difference system $D_{q}Yx=AYqx$ with the initial condition $Y0=Y_0$ where $A$ is a constant square complex matrix, $D_q$ is the Jackson $q$-derivative and $0<q<1$, is asymptotically stable if and only if $\Re \lambda <0$ for all $\lambda \in \sigma A$ where $\sigma A$ is the set of all eigenvalues of $A$ (the spectrum of $A$). This results are exploited to provide the orthogonality property of the discrete $q$-Hermite matrix polynomials.

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1. Introduction

The beginning of $q$-polynomials is related to the work of Rogers and Ramanujan of the late 19th and early 20th century, see [1, 2]. This area of mathematics has grown into an interesting level due to the connection work between Rogers–Ramanujan identities and certain families of orthogonal $q$-polynomials such as $q$-Hermite and $q$-Ultraspherical which have been introduced by Andrews, Askey, Ismail, and Bressoud [3, 4]. The classical orthogonal polynomials have many applications in the theory of linear control systems, in robotics and computer aided geometric design [5, 6].

Hermite–Bell’s Polynomials for Negative Powers are introduced and studied by [7] and some fundamental analytical properties to Hermite polynomials are investigated in [8]. Several classical families of orthogonal polynomials (Laguerre, Hermite, Gegenbauer, etc.) have been extended to the matrix framework, see [9–11]. As a matter in fact, the Hermit matrix polynomial has received a special interest in the literature. An explicit expression, orthogonality property, Rodrigues formula, and other properties have been given in [12–14].

As a first step to extend the matrix framework of quantum calculus, Salem introduced and studied the $q$-gamma and $q$-beta matrix functions [15], the basic Gauss hypergeometric function [16] and the $q$-Laguerre matrix polynomials [17]. He also presented in [18] the discrete $q$-Hermite matrix polynomials of type I $H_{n}x,A;q$ by means of the generating matrix function$$\begin{array}{}\text{(1)}& {t;q}_{\infty }{-t;q}_{\infty }{xt\sqrt{A};q}_{\infty }^{-1}={\displaystyle \sum _{n=0}^{\infty }\frac{H_{n}x,A;q}{{q;q}_n}}{t}^n,\end{array}$$where ${A;q}_{\infty }$ is the $q$-shifted factorial matrix function defined [15] as follows:$$\begin{array}{c}\text{(2)}& {A;q}_{\infty }=\underset{n⟶\infty }{\text{lim}}{A;q}_n=\underset{n⟶\infty }{\text{lim}}{\displaystyle \prod _{k=0}^{n-1}I-A{q}^k},\end{array}$$and $A$ is a square matrix satisfying the following conditions:$$\begin{array}{}\text{(3)}& \Re z>0,\hspace{1em}\text{for\hspace{0.17em}all}z\in \sigma A,\end{array}$$where $\sigma A$ denotes the set of all eigenvalues of $A$, and$$\begin{array}{}\text{(4)}& A<1\hspace{1em}\text{and\hspace{0.17em}}{q}^{-m}\notin \sigma A,\hspace{0.17em}m\in {\mathbb{N}}_0,\end{array}$$for the operator norm $\cdot$ corresponding to the 2-norm. The discrete $q$-Hermite matrix polynomials has an explicit representation of the form.$$\begin{array}{}\text{(5)}& H_{n}x,A;q={\displaystyle \sum _{k=0}^{n/2}\frac{{-1}^k{q}^{kk-1}{q;q}_n}{{q^2;q^2}_k{q;q}_{n-2k}}{x\sqrt{A}}^{n-2k}},\hspace{1em}n\in {\mathbb{N}}_0,\end{array}$$where $\sqrt{A}$ is the square root of the matrix $A$ in the sense of the matrix functional calculus $\sqrt{A}=\exp 1/2\log A$, and $\log$ denotes the principal branch of the complex logarithm, and satisfies the three terms recurrence relation.$$\begin{array}{}\text{(6)}& H_{n}x,A;q=x\sqrt{A}{H}_{n-1}x,A;q-q^{n-2}1-q^{n-1}{H}_{n-2}x,A;q,\hspace{1em}n\in \mathbb{N},\end{array}$$and $q$-difference matrix equation.$$\begin{array}{}\text{(7)}& D_q^2-\frac{q^{2-n}}{1-q}xA{D}_q+\frac{q^{2-n}}{1-q}{n}_{q}A{H}_{n}x,A;q=0,\end{array}$$where $D_{q}f$ is the $q$-derivative of a function $f$ defined as follows:$$\begin{array}{}\text{(8)}& D_{q}fx=\frac{fx-fxq}{1-qx},q\ne 1,\hspace{1em}x\ne 0.\end{array}$$

An easy consequence of the above definition, the $q$-derivative of two product functions is$$\begin{array}{c}\text{(9)}& D_{q}fxgx=gx{D}_{q}fx+fqx{D}_{q}gx=fx{D}_{q}gx+gqx{D}_{q}fx.\end{array}$$

The discrete $q$-Hermite matrix polynomials of type II $h_{n}x,A;q$ was introduced in [18] which is related to the discrete $q$-Hermite matrix polynomials of type I defined by (5) in the following way:$$\begin{array}{}\text{(10)}& h_{n}x,A;q={-i}^n{H}_{n}ix,A;\frac{1}{q}.\end{array}$$

The main goal of this paper is to investigate the orthogonality property of the discrete $q$-Hermite matrix polynomials of type I.

2. Asymptotically Stable Solution to Autonomous $q$-Difference System

The stability theory of differential and difference equations is of main interest in physical systems. The complete controllability and observability of $q$-difference linear control systems of first order have been investigated by Bangerezako [19]. One of the linear systems considered in [19] is the autonomous $q$-difference system.$$\begin{array}{c}\text{(11)}& D_{q}Yx=AYqx,Y0=Y_0,\end{array}$$where $A$ is constant matrix in ${ℂ}^{n\times n}$ and $0<q<1$. It has been proven in [19] that the solution of the autonomous $q$-difference system (11) is unique and has the following form:$$\begin{array}{c}\text{(12)}& Yx=Y_0{\displaystyle \prod _{n=0}^{\infty }I+1-qxA{q}^n}=Y_0{E}_{q}xA,\end{array}$$where $E_{q}A$ is the $q$-exponential matrix function and also can be defined as follows:$$\begin{array}{}\text{(13)}& E_{q}A={\displaystyle \sum _{n=0}^{\infty }\frac{q^{\begin{array}{c}n\\ 2\end{array}}}{n_q!}}{A}^n,\end{array}$$where $a_q$ and $n_q!$ are the basic number and the basic factorial function defined, respectively, as follows:$$\begin{array}{}\text{(14)}& a_q=\frac{1-q^a}{1-q},\hspace{1em}q\ne 1,\end{array}$$and$$\begin{array}{}\text{(15)}& n_q!=n_q{n-1}_q\cdots 2_q{1}_q,\hspace{1em}n\in \mathbb{N}\text{with}{0}_q!=1.\end{array}$$

Definition 1.

The solution of the autonomous system (11) is said to be stable if there exists $\epsilon >0$ such that $Yx<\epsilon$ for all $x>0$. The solution is said to be asymptotically stable if, in addition to being stable, $Yx⟶0$ as $x⟶\infty$.

Definition 2.

The $q$-gamma function is defined as follows:$$\begin{array}{c}\text{(16)}& {\Gamma }_q\alpha ={\displaystyle {\int }_0^{1/1-q}{t}^{\alpha -1}{E}_q-qt{\text{d}}_{q}t}={\displaystyle {\int }_0^{\infty /1-q}{t}^{\alpha -1}{E}_q-qt{\text{d}}_{q}t},\hspace{1em}\Re \alpha >0,\end{array}$$and has the identity ${\mathrm{\Gamma }}_q\alpha +1={\alpha }_q{\mathrm{\Gamma }}_q\alpha$, where the $q$-integration of Jackson in a generic interval $a,b$ is given as follows:$$\begin{array}{}\text{(17)}& {\displaystyle {\int }_a^{b}ft{\text{d}}_{q}t}={\displaystyle {\int }_0^{b}ft{\text{d}}_{q}t}-{\displaystyle {\int }_0^{a}ft{\text{d}}_{q}t},\end{array}$$where$$\begin{array}{}\text{(18)}& {\displaystyle {\int }_0^{a}f}x{d}_{q}x=a1-q{\displaystyle \sum _{n=0}^{\infty }f}a{q}^n{q}^n,\end{array}$$provided the sum converges absolutely and the improper integral is defined in the following way (see [20, 21]):$$\begin{array}{}\text{(19)}& {\displaystyle {\int }_0^{\infty /1-q}fx{\text{d}}_{q}x}={\displaystyle \sum _{n=-\infty }^{\infty }{q}^{n}f\frac{q^n}{1-q}}.\end{array}$$

The bilateral $q$-integral is defined as follows:$$\begin{array}{}\text{(20)}& {\displaystyle {\int }_{-\infty }^{\infty }fx{\text{d}}_{q}x}=1-q{\displaystyle \sum _{n=0}^{\infty }{q}^{n}f{q}^n+f-q^n}.\end{array}$$

The $q$-integrating by parts is given for suitable functions $f$ and $g$ by$$\begin{array}{}\text{(21)}& {\displaystyle {\int }_a^{b}fx{D}_{q}gx{\text{d}}_{q}x}=fbgb-faga-{\displaystyle {\int }_a^{b}gqx{D}_{q}fx{\text{d}}_{q}x}.\end{array}$$

Let the two series$$\begin{array}{c}\text{(22)}& Sux={\displaystyle \sum _{n=0}^{\infty }\frac{1-q{q}^n}{1+ux1-q{q}^n}},Tx=i{\displaystyle \sum _{k=0}^{\infty }\frac{v1-q{q}^k}{1+ux1-q{q}^{k}1+\lambda x1-q{q}^k}},\end{array}$$be defined for large $x$ and $0<q<1$ where $\lambda =u+iv\in ℂ$ and $u,v\in ℝ$. It is clear that$$\begin{array}{}\text{(23)}& \underset{x⟶\infty }{\text{lim}}\frac{Tx}{Sux}=\underset{x⟶\infty }{\text{lim}}{\displaystyle \sum _{k=0}^{\infty }\frac{iv{q}^k}{{\displaystyle {\sum }_{n=0}^{\infty }{q}^{n}1+ux1-q{q}^{k}1+\lambda x1-q{q}^k/1+ux1-q{q}^n}}}=0,\text{(24)}& \underset{x⟶\infty }{\text{lim}}\frac{1}{xS\lambda x}=\underset{x⟶\infty }{\text{lim}}\frac{1}{{\displaystyle {\sum }_{n=0}^{\infty }x1-q{q}^n/1+\lambda x1-q{q}^n}}=0.\end{array}$$

The interchange of the limits and the series has verified due to the convergence of the series.

Lemma 1.

For all complex number $\lambda$, we have$$\begin{array}{}\text{(25)}& \underset{x⟶\infty }{\text{lim}}{E}_q\lambda x=0,0<q<1,\end{array}$$if and only if $\Re \lambda <0$.

Proof.

From the definitions of $q$-gamma function (16), we get$$\begin{array}{c}\text{(26)}& 1={\Gamma }_{q}1=-{\displaystyle {\int }_0^{\infty /1-q}{D}_q{E}_q^{-t}{\text{d}}_{q}t}=1-{\text{lim}}_{y⟶\infty }{E}_q\frac{-y}{1-q},\end{array}$$which reveals that ${\text{lim}}_{y⟶\infty }{E}_q-y/1-q=0$ or equivalently, ${\text{lim}}_{x⟶\infty }{E}_{q}ux=0$ if $u<0$. Now, we can write$$\begin{array}{c}\text{(27)}& E_q\lambda x={\displaystyle \prod _{n=0}^{\infty }1+\lambda x1-q{q}^n}={\displaystyle \prod _{n=0}^{\infty }1+ux1-q{q}^n}{\displaystyle \prod _{n=0}^{\infty }1+\frac{ivx1-q{q}^n}{1+ux1-q{q}^n}}\\ =E_{q}ux{\displaystyle \prod _{n=0}^{\infty }1+\frac{ivx1-q{q}^n}{1+ux1-q{q}^n}},\lambda =u+iv.\end{array}$$

Using L’Hospital rule gives$$\begin{array}{}\text{(28)}& {\text{lim}}_{x⟶\infty }{E}_q\lambda x1+\frac{1}{u}\underset{x⟶\infty }{\text{lim}}\frac{Tx}{Sux}=0.\end{array}$$

In view of (23), we get $\underset{x⟶\infty }{\text{lim}}{E}_q\lambda x=0$ if $u=\Re \lambda <0$. Now, let $u\ge 0$, then $\underset{x⟶\infty }{\text{lim}}{E}_{q}ux\ne 0$ which reveals that $\underset{x⟶\infty }{\text{lim}}{E}_q\lambda x\ne 0$.

Lemma 2.

For all complex numbers $\alpha$ and $\lambda$, we have$$\begin{array}{}\text{(29)}& \underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x=0,\end{array}$$if and only if $\Re \lambda <0$ and $\Re \alpha \ge 0$ or $\Re \lambda \le 0$ and $\Re \alpha <0$.

Proof.

There are nine cases that can be rewritten in five cases

Case 1.

$\Re \alpha \le 0$ and $\Re \lambda <0$,$$\begin{array}{}\text{(30)}& \underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x=\underset{x⟶\infty }{\text{lim}}{x}^{\alpha }\underset{x⟶\infty }{\text{lim}}{E}_q\lambda x=0.\end{array}$$

Case 2.

$\Re \alpha >0$ and $\Re \lambda <0$, using L’Hospital rule gives$$\begin{array}{}\text{(31)}& \underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x1+\frac{\alpha }{\lambda }\underset{x⟶\infty }{\text{lim}}\frac{1}{xS\lambda x}=0.\end{array}$$

In view of (24), we get $\underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x=0$.

Case 3.

$\Re \alpha <0$ and $\Re \lambda =0$, if $\Im \lambda =0$, then $E_{q}0=1$ and so$$\begin{array}{}\text{(32)}& \underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x=\underset{x⟶\infty }{\text{lim}}{x}^{\alpha }=0.\end{array}$$

If $\Im \lambda \ne 0$, we find that$$\begin{array}{}\text{(33)}& \underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x\le \underset{x⟶\infty }{\text{lim}}{x}^{\Re \alpha }{\displaystyle \prod _{n=0}^{\infty }1+vx{q}^n},v=\Im \lambda .\end{array}$$

The product above approaches infinity as $x⟶\infty$ and so we can use the L’Hospital rule to obtain$$\begin{array}{}\text{(34)}& {\text{lim}}_{x⟶\infty }{x}^{\Re \alpha }{\displaystyle \prod _{n=0}^{\infty }1+vx{q}^n}1+\frac{\Re \alpha }{v}{\text{lim}}_{x⟶\infty }\frac{1}{xSvx}=0,\end{array}$$which reveals by using (24) that $\underset{x⟶\infty }{\text{lim}}{x}^{\Re \alpha }{\displaystyle {\prod }_{n=0}^{\infty }1+vx{q}^n}=0$ and so $\underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x=0$.

Case 4.

$\Re \alpha \ge 0$ and $\Re \lambda \ge 0$, then $\underset{x⟶\infty }{\text{lim}}{E}_q\lambda x\ne 0$ which yields that $\underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x\ne 0$.

Case 5.

$\Re \alpha <0$ and $\Re \lambda >0$, let $-k-1\le \Re \alpha <-k$ for some positive integer $k$, then we have$$\begin{array}{}\text{(35)}& {\text{lim}}_{x⟶\infty }{x}^{\alpha }{E}_q\lambda x={\text{lim}}_{x⟶\infty }{\displaystyle \sum _{n=0}^k\frac{q^{\begin{array}{c}n\\ 2\end{array}}{\lambda }^n}{n_q!}}{x}^{\alpha +n}+{\text{lim}}_{x⟶\infty }{\displaystyle \sum _{n=k+1}^{\infty }\frac{q^{\begin{array}{c}n\\ 2\end{array}}{\lambda }^n}{n_q!}}{x}^{\alpha +n}.\end{array}$$

It is obvious that the first limit equals zero and the last limit does not equal zero which means that $\underset{x⟶\infty }{\text{lim}}{x}^{\alpha }{E}_q\lambda x\ne 0$.

These end the proof.

Lemma 3.

For all complex number $\lambda$, we have$$\begin{array}{}\text{(36)}& {{\text{lim}}_{x⟶\infty }\frac{d^n}{d{z}^n}{E}_{q}xz|}_{z=\lambda }=0,n\in {\mathbb{N}}_0,\end{array}$$if and only if $\Re \lambda <0$.

Proof.

The case of $n=0$ is proven in the lemma above. When $n=1$, the logarithmic derivative for $E_{q}xz$ with respect to $z$ gives$$\begin{array}{}\text{(37)}& \frac{d}{dz}{E}_{q}xz=x{E}_{q}xzSxz.\end{array}$$

Iterating this process yields$$\begin{array}{}\text{(38)}& \frac{d^n}{d{z}^n}{E}_{q}xz=x^n{E}_{q}xzgxz,\end{array}$$where $gxz$ is a finite sum of the infinite series ${{\displaystyle {\sum }_{k=0}^{\infty }{1-q{q}^k/1+1-qxz{q}^k}^r}}^s$ and $r,s\in \mathrm{1,2},\dots ,n$ which tend to zero as $x⟶\infty$. By virtue of the results obtained in the lemma above, we get$$\begin{array}{}\text{(39)}& {{\text{lim}}_{x⟶\infty }\frac{d^n}{d{z}^n}{E}_{q}xz|}_{z=\lambda }={\text{lim}}_{x⟶\infty }{x}^n{E}_q\lambda x{\text{lim}}_{x⟶\infty }g\lambda x=0,\Re \lambda <0.\end{array}$$

This ends the proof.

Theorem 1.

The solution of the autonomous $q$-difference system (11) is asymptotically stable if and only if $\alpha A<0$ for all $x>0$ where $\alpha A$ is the spectral abscissa defined as follows:$$\begin{array}{}\text{(40)}& \alpha A=\max \Re z:z\in \sigma A.\end{array}$$

Proof.

According to the Jordan matrix decomposition [22], with regard to the square matrix $A\in {ℂ}^{n\times n}$, there exists an invertible matrix $P$ such that$$\begin{array}{c}\text{(41)}& A=PJ{P}^{-1}=P\text{diag}{J}_1,J_2,\dots ,J_k{P}^{-1},\end{array}$$where the $J_i$ are the Jordan blocks of $A$ with the eigenvalues of $A$ on the diagonal. The Jordan blocks are uniquely determined by $A$ and have the form$$\begin{array}{}\text{(42)}& J_i={\begin{array}{cccccc}{\lambda }_i& 1& & & & \\ & {\lambda }_i& 1& & & \\ & & \ddots & \ddots & & \\ & & & {\lambda }_i& 1& \\ & & & & {\lambda }_i& \end{array}}_{n_i\times n_i},i=\mathrm{1,2},\dots k,\end{array}$$where ${\lambda }_i\in \sigma A$ and ${\displaystyle {\sum }_{i=1}^k{n}_i}=n$. From the definition of $q$-exponential matrix function (13) and the relation (41), we get$$\begin{array}{c}\text{(43)}& E_{q}xA=P{\displaystyle \sum _{m=0}^{\infty }\frac{q^{\begin{array}{c}m\\ 2\end{array}}\text{diag}{J}_1^m,J_2^m,\dots ,J_k^m{x}^m}{m_q!}}{P}^{-1}=P\text{diag}{E}_{q}x{J}_1,E_{q}x{J}_2,\dots ,E_{q}x{J}_k{P}^{-1},\end{array}$$where the matrix $E_{q}x{J}_i$ can be written in the form$$\begin{array}{}\text{(44)}& E_{q}x{J}_i={\begin{array}{cccccc}f\lambda & f^{\prime }\lambda & \frac{f^{″}\lambda }{2}& \dots & \frac{f^{n_k-2}\lambda }{n_k-2!}& \frac{f^{n_k-1}\lambda }{n_k-1!}\\ 0& f\lambda & f^{\prime }\lambda & \cdots & \frac{f^{n_k-3}\lambda }{n_k-3!}& \frac{f^{n_k-2}\lambda }{n_k-2!}\\ 0& 0& f\lambda & \cdots & \frac{f^{n_k-4}\lambda }{n_k-4!}& \frac{f^{n_k-3}\lambda }{n_k-3!}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & f\lambda & f^{\prime }\lambda \\ 0& 0& 0& \cdots & 0& f\lambda \end{array}}_{\lambda ={\lambda }_i},\end{array}$$and $f\lambda =E_{q}x\lambda$. In view of the results obtained by Lemma 3, the limit of nonzero elements of $E_{q}x{J}_i$ can be computed as follows:$$\begin{array}{}\text{(45)}& {\text{lim}}_{x⟶\infty }\frac{d^r}{d{\lambda }^r}{E_{q}x\lambda |}_{\lambda ={\lambda }_i}=0,\hspace{1em}r=\mathrm{0,1},\dots n_k-2,\Re {\lambda }_i<0,\end{array}$$which concludes that$$\begin{array}{}\text{(46)}& {\text{lim}}_{x⟶\infty }\mathrm{\Vert }{E}_{q}xA\mathrm{\Vert }=0,\Re {\lambda }_i<0,\hspace{1em}{\lambda }_i\in \sigma A.\end{array}$$

This completes the proof.

3. Orthogonality Property

Suppose that the inner product $f,g$ for a suitable two matrix-valued functions $f$ and $g$ is defined as follows:$$\begin{array}{}\text{(47)}& fx,gx={\displaystyle {\int }_{-\infty }^{\infty }wx,A;qfxgx{\text{d}}_{q}x},\end{array}$$where the weight function $wx,A;q$ is defined as follows:$$\begin{array}{c}\text{(48)}& wx,A;q={qx\sqrt{A};q}_{\infty }{-qx\sqrt{A};q}_{\infty }={q^2{x}^{2}A;q^2}_{\infty }.\end{array}$$

Definition 3.

Let $P_{n}x$ be a matrix polynomials for integer $n\ge 0$. We say that the sequence ${P_{n}x}_{n\ge 0}$ is an orthogonal matrix polynomials sequence with respect to the inner product $\cdot ,\cdot$ provided for all nonnegative integers $n$ and $m$.

(1) $P_{n}x$ is a matrix polynomial of degree $n$ with nonsingular leading coefficient.

(2) $P_{n}x,P_{m}x=0$ for all $n\ne m$.

(3) $P_{n}x,P_{n}x$ is invertible for $n\ge 0$.

Theorem 2.

Assume that $n,m\in {\mathbb{N}}_0$ and $A$ is a square complex matrix satisfying the conditions (3) and (4). Then, we have$$\begin{array}{}\text{(49)}& H_{n}x,A;q,H_{m}x,A;q=0,n\ne m.\end{array}$$

Proof.

According to the $q$-derivative of two product functions (9), we get$$\begin{array}{c}\text{(50)}& D_{q}wx,A;q{D}_q{H}_{n}x,A;q=wqx,A;q{D}_q^2{H}_{n}x,A;q+D_{q}wx,A;q{D}_q{H}_{n}x,A;q.\end{array}$$

From the $q$-difference matrix (7), we obtain$$\begin{array}{c}\text{(51)}& D_{q}wx,A;q{D}_q{H}_{n}x,A;q=q^{2-n}{n}_{q}wqx,A;qAx{D}_q{H}_{n}x,A;q-\frac{H_{n}x,A;q}{1-q}=q^{2-n}{n}_{q}wqx,A;qA\frac{H_{n}x,A;q-H_{n}qx,A;q}{1-q}-\frac{H_{n}x,A;q}{1-q}\\ ={\lambda }_{n}wqx,A;q{H}_{n}qx,A;q,n\in {\mathbb{N}}_0,\end{array}$$where$$\begin{array}{}\text{(52)}& {\lambda }_n=-\frac{q^{2-n}}{1-q}{n}_{q}A.\end{array}$$

Also, for all integer $m\in {\mathbb{N}}_0$, we have$$\begin{array}{}\text{(53)}& D_{q}wx,A;q{D}_q{H}_{m}x,A;q={\lambda }_{m}wqx,A;q{H}_{m}qx,A;q.\end{array}$$

By multiplying (51) by $H_{m}qx,A;q$ from the right and (53) by $H_{n}qx,A;q$ from the left followed by subtraction and $q$-integration from $-\infty$ to $\infty$, yields$$\begin{array}{c}\text{(54)}& {\lambda }_n-{\lambda }_m{H}_{n}x,A;q,H_{m}x,A;q={\lambda }_n-{\lambda }_{m}q{\displaystyle {\int }_{-\infty }^{\infty }wqx,A;q{H}_{n}qx,A;q{H}_{m}qx,A;q{\text{d}}_{q}x}=q{\displaystyle {\int }_{-\infty }^{\infty }{D}_{q}wx,A;q{D}_q{H}_{n}x,A;q}{H}_{m}qx,A;q{\text{d}}_{q}x\\ -q{\displaystyle {\int }_{-\infty }^{\infty }{H}_{n}qx,A;q{D}_{q}wx,A;q{D}_q{H}_{m}x,A;q{\text{d}}_{q}x}.\end{array}$$

On $q$-integrating by parts gives$$\begin{array}{c}\text{(55)}& {\lambda }_n-{\lambda }_m{H}_{n}x,A;q,H_{m}x,A;q={qwx,A;q{D}_q{H}_{n}x,A;q{H}_{m}x,A;q|}_{-\infty }^{\infty }{-qwx,A;q{D}_q{H}_{m}x,A;q{H}_{n}x,A;q|}_{-\infty }^{\infty }.\end{array}$$

It was shown in Theorem 1 that $E_{q}xA⟶0$ as $x⟶\infty$ if and only if $\alpha A<0$ which is equivalent to $E_q-xA={1-qxA;q}_{\infty }⟶0$ as $x⟶\infty$ if and only if $\beta A>0$ where$$\begin{array}{c}\text{(56)}& \beta A=-\alpha -A=\min \Re z:z\in \sigma A.\end{array}$$

This concludes that$$\begin{array}{}\text{(57)}& {\text{lim}}_{x⟶\infty }wx,A;q={\text{lim}}_{x⟶\infty }{q^2{x}^{2}A;q^2}_{\infty }=0,\end{array}$$if and only if $\beta A>0$. Hence, the proof is complete.

Lemma 4.

Let $A$ be a square complex matrix satisfying the conditions (3) and (4). Then, we have$$\begin{array}{}\text{(58)}& {\displaystyle {\int }_{-\infty }^{\infty }{q^2{x}^{2}A;q^2}_{\infty }{\text{d}}_{q}x}=1-q{q;q}_{\infty }{-q;q}_{\infty }{-1;q}_{\infty }{\sqrt{A}}^{-1}.\end{array}$$

Proof.

Since the function ${q^2{x}^{2}A;q^2}_{\infty }$ is even, we obtain$$\begin{array}{c}\text{(59)}& {\displaystyle {\int }_{-\infty }^{\infty }{q^2{x}^{2}A;q^2}_{\infty }{\text{d}}_{q}x}=2{\displaystyle {\int }_0^{\infty }{q^2{x}^{2}A;q^2}_{\infty }{\text{d}}_{q}x}=2{\displaystyle {\int }_0^{\infty }{-qx\sqrt{A};q}_{\infty }{qx\sqrt{A};q}_{\infty }{\text{d}}_{\text{q}}x}\\ =2{\displaystyle {\int }_0^{\infty }{E}_q\frac{qx\sqrt{A}}{1-q}{E}_q-\frac{qx\sqrt{A}}{1-q}{\text{d}}_{q}x}.\end{array}$$

From (13), we get$$\begin{array}{}\text{(60)}& {\displaystyle {\int }_{-\infty }^{\infty }{q^2{x}^{2}A;q^2}_{\infty }{\text{d}}_{q}x}=2{\displaystyle \sum _{n=0}^{\infty }\frac{q^{nn+1/2}}{{q;q}_n}}{\sqrt{A}}^n{\displaystyle {\int }_0^{\infty }{x}^n{E}_q-\frac{qx\sqrt{A}}{1-q}{\text{d}}_{q}x}.\end{array}$$

The interchange of the summation and integration is justified due to their convergence.

It has been shown in [23] that if $\sigma A\cap -\infty ,0=\varphi$, then $A$ has a unique square root with $\sigma \sqrt{A}$ in the open right (complex) half plane. Now, replacing $x$ by $1-qt{\sqrt{A}}^{-1}$, according to the former notice, we find that$$\begin{array}{}\text{(61)}& {\displaystyle {\int }_{-\infty }^{\infty }{q^2{x}^{2}A;q^2}_{\infty }{\text{d}}_{q}x}=21-q{\sqrt{A}}^{-1}{\displaystyle \sum _{n=0}^{\infty }\frac{q^{nn+1/2}}{n_q!}}{\displaystyle {\int }_0^{\infty /1-q}{t}^n{E}_q-qt{\text{d}}_{q}t}.\end{array}$$

In view of the definition of $q$-gamma function (16), the previous integral equals ${\mathrm{\Gamma }}_{q}n+1=n_q!$. Therefore,$$\begin{array}{}\text{(62)}& {\displaystyle {\int }_{-\infty }^{\infty }{q^2{x}^{2}A;q^2}_{\infty }{\text{d}}_{q}x}=21-q{\sqrt{A}}^{-1}{\displaystyle \sum _{n=0}^{\infty }{q}^{nn+1/2}}.\end{array}$$

The Jacobi triple product can be written as [21].$$\begin{array}{}\text{(63)}& {\displaystyle \sum _{n=-\infty }^{\infty }{-1}^n}{q}^{\begin{array}{c}n\\ 2\end{array}}{x}^n={q;q}_{\infty }{x;q}_{\infty }{q{x}^{-1};q}_{\infty }.\end{array}$$

Replacing $x$ by $-q$ with noting that$$\begin{array}{}\text{(64)}& {\displaystyle \sum _{n=-\infty }^{\infty }{q}^{nn+1/2}}=2{\displaystyle \sum _{n=0}^{\infty }{q}^{nn+1/2}},\end{array}$$gives$$\begin{array}{}\text{(65)}& {\displaystyle \sum _{n=0}^{\infty }{q}^{nn+1/2}}=\frac{1}{2}{q;q}_{\infty }{-q;q}_{\infty }{-1;q}_{\infty }.\end{array}$$

Substituting into (62) to obtain the desired result.

Theorem 3.

Assume that $n\in {\mathbb{N}}_0$ and $A$ is a square complex matrix satisfying the conditions (3) and (4). Then, we have$$\begin{array}{}\text{(66)}& H_{n}x,A;q,H_{n}x,A;q=1-q{q;q}_n{q,-q,-1;q}_{\infty }{q}^{\begin{array}{c}n\\ 2\end{array}}{\sqrt{A}}^{2n-1}.\end{array}$$

Proof.

The Rodrigues-type formula for the discrete $q$-Hermite matrix polynomials was derived by the following equation [18]:$$\begin{array}{}\text{(67)}& wx,A;q{H}_{n}x,A;q=\xi n{D}_{q^{-1}}^{n}wx,A;q,n\in {\mathbb{N}}_0,\end{array}$$where$$\begin{array}{}\text{(68)}& \xi n={-1}^n{1-q}^n{q}^{nn-3/2}{\sqrt{A}}^n.\end{array}$$

Therefore, (47) gives$$\begin{array}{c}\text{(69)}& H_{n}x,A;q,H_{n}x,A;q={\displaystyle {\int }_{-\infty }^{\infty }wx,A;q{H}_{n}x,A;q{H}_{n}x,A;q{\text{d}}_{q}x}=\xi n{\displaystyle {\int }_{-\infty }^{\infty }{D}_{q^{-1}}^{n}wx,A;q{H}_{n}x,A;q{\text{d}}_{q}x}.\end{array}$$

It is easy, from the definition of $q$-derivative (8), to see that$$\begin{array}{}\text{(70)}& D_{q^{-1}}fx=q{D}_{q}fx{q}^{-1}.\end{array}$$

Using the identity with putting$$\begin{array}{c}\text{(71)}& fx=D_{q^{-1}}^{n-1}wx,A;q={\xi }^{-1}n-1wx,A;q{H}_{n-1}x,A;q,\end{array}$$to obtain$$\begin{array}{}\text{(72)}& H_{n}x,A;q,H_{n}x,A;q=q\xi n{\displaystyle {\int }_{-\infty }^{\infty }{D}_{q}fx{q}^{-1}{H}_{n}x,A;q{\text{d}}_{q}x}.\end{array}$$

On $q$-integrating by parts yields$$\begin{array}{}\text{(73)}& H_{n}x,A;q,H_{n}x,A;q=q\xi n{fx{q}^{-1}{H}_{n}x,A;q|}_{-\infty }^{\infty }-{\displaystyle {\int }_{-\infty }^{\infty }fx{D}_q{H}_{n}x,A;q{\text{d}}_{q}x}.\end{array}$$

By virtue of the results obtained in Theorem 1, the first term equals zero and by using the identity [18].$$\begin{array}{}\text{(74)}& D_q{H}_{n}x,A;q=n_q\sqrt{A}{H}_{n-1}x,A;q,\end{array}$$we can arrive at$$\begin{array}{}\text{(75)}& H_{n}x,A;q,H_{n}x,A;q=q^{n-1}1-q^{n}A{H}_{n-1}x,A;q,H_{n-1}x,A;q.\end{array}$$

Iterating this process yields$$\begin{array}{}\text{(76)}& H_{n}x,A;q,H_{n}x,A;q=q^{nn-1/2}{q;q}_n{A}^n{H}_{0}x,A;q,H_{0}x,A;q.\end{array}$$

From the explicit formula (5) for the discrete $q$-Hermite matrix polynomials at $n=0$, we get $H_{0}x,A;q=I$ and so from Lemma 4, we get$$\begin{array}{c}\text{(77)}& H_{0}x,A;q,H_{0}x,A;q={\displaystyle {\int }_{-\infty }^{\infty }{q^2{x}^{2}A;q^2}_{\infty }{\text{d}}_{q}x}=1-q{q;q}_{\infty }{-q;q}_{\infty }{-1;q}_{\infty }{\sqrt{A}}^{-1}.\end{array}$$

Substituting into (76) to obtain the desired result.

Summarization of the results obtained in this section can be stated in the following theorem:

Theorem 4.

Assume that $n\in {\mathbb{N}}_0$ and $A$ be a square complex matrix satisfying the conditions (3) and (4). Then, the discrete $q$-Hermite matrix polynomial is an orthogonal polynomial with respect to the inner product $\cdot ,\cdot$ defined in (47). Furthermore,$$\begin{array}{}\text{(78)}& H_{n}x,A;q,H_{m}x,A;q=1-q{q;q}_n{q,-q,-1;q}_{\infty }{q}^{\begin{array}{c}n\\ 2\end{array}}{\sqrt{A}}^{2n-1}{\delta }_{nm},\end{array}$$where ${\delta }_{nm}$ is the Kronecker delta.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under Grant no. KEP-PhD-57-130-42. The authors, therefore, acknowledge with thanks DSR technical and financial support.

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