Orthogonality Property of the Discrete -Hermite Matrix Polynomials

Abstract

In this paper, we prove that the solution of the autonomous -difference system with the initial condition where is a constant square complex matrix, is the Jackson -derivative and , is asymptotically stable if and only if for all where is the set of all eigenvalues of (the spectrum of ). This results are exploited to provide the orthogonality property of the discrete -Hermite matrix polynomials.

1. Introduction

The beginning of -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 -polynomials such as -Hermite and -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 -gamma and -beta matrix functions [15], the basic Gauss hypergeometric function [16] and the -Laguerre matrix polynomials [17]. He also presented in [18] the discrete -Hermite matrix polynomials of type I by means of the generating matrix functionwhere is the -shifted factorial matrix function defined [15] as follows:and is a square matrix satisfying the following conditions:where denotes the set of all eigenvalues of , andfor the operator norm corresponding to the 2-norm. The discrete -Hermite matrix polynomials has an explicit representation of the form.where is the square root of the matrix in the sense of the matrix functional calculus , and denotes the principal branch of the complex logarithm, and satisfies the three terms recurrence relation.and -difference matrix equation.where is the -derivative of a function defined as follows:

An easy consequence of the above definition, the -derivative of two product functions is

The discrete -Hermite matrix polynomials of type II was introduced in [18] which is related to the discrete -Hermite matrix polynomials of type I defined by (5) in the following way:

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

2. Asymptotically Stable Solution to Autonomous -Difference System

The stability theory of differential and difference equations is of main interest in physical systems. The complete controllability and observability of -difference linear control systems of first order have been investigated by Bangerezako [19]. One of the linear systems considered in [19] is the autonomous -difference system.where is constant matrix in and . It has been proven in [19] that the solution of the autonomous -difference system (11) is unique and has the following form:where is the -exponential matrix function and also can be defined as follows:where and are the basic number and the basic factorial function defined, respectively, as follows:and

Definition 1. The solution of the autonomous system (11) is said to be stable if there exists such that for all . The solution is said to be asymptotically stable if, in addition to being stable, as .

Definition 2. The -gamma function is defined as follows:and has the identity , where the -integration of Jackson in a generic interval is given as follows:whereprovided the sum converges absolutely and the improper integral is defined in the following way (see [20, 21]):The bilateral -integral is defined as follows:The -integrating by parts is given for suitable functions and byLet the two seriesbe defined for large and where and . It is clear thatThe interchange of the limits and the series has verified due to the convergence of the series.

Lemma 1. For all complex number , we haveif and only if .

Proof. From the definitions of -gamma function (16), we getwhich reveals that or equivalently, if . Now, we can writeUsing L’Hospital rule givesIn view of (23), we get if . Now, let , then which reveals that .

Lemma 2. For all complex numbers and , we haveif and only if and or and .

Proof. There are nine cases that can be rewritten in five cases

Case 1. and ,

Case 2. and , using L’Hospital rule givesIn view of (24), we get .

Case 3. and , if , then and soIf , we find thatThe product above approaches infinity as and so we can use the L’Hospital rule to obtainwhich reveals by using (24) that and so .

Case 4. and , then which yields that .

Case 5. and , let for some positive integer , then we haveIt is obvious that the first limit equals zero and the last limit does not equal zero which means that .
These end the proof.

Lemma 3. For all complex number , we haveif and only if .

Proof. The case of is proven in the lemma above. When , the logarithmic derivative for with respect to givesIterating this process yieldswhere is a finite sum of the infinite series and which tend to zero as . By virtue of the results obtained in the lemma above, we getThis ends the proof.

Theorem 1. The solution of the autonomous -difference system (11) is asymptotically stable if and only if for all where is the spectral abscissa defined as follows:

Proof. According to the Jordan matrix decomposition [22], with regard to the square matrix , there exists an invertible matrix such thatwhere the are the Jordan blocks of with the eigenvalues of on the diagonal. The Jordan blocks are uniquely determined by and have the formwhere and . From the definition of -exponential matrix function (13) and the relation (41), we getwhere the matrix can be written in the formand . In view of the results obtained by Lemma 3, the limit of nonzero elements of can be computed as follows:which concludes thatThis completes the proof.

3. Orthogonality Property

Suppose that the inner product for a suitable two matrix-valued functions and is defined as follows:where the weight function is defined as follows:

Definition 3. Let be a matrix polynomials for integer . We say that the sequence is an orthogonal matrix polynomials sequence with respect to the inner product provided for all nonnegative integers and .(1) is a matrix polynomial of degree with nonsingular leading coefficient.(2) for all .(3) is invertible for .

Theorem 2. Assume that and is a square complex matrix satisfying the conditions (3) and (4). Then, we have

Proof. According to the -derivative of two product functions (9), we getFrom the -difference matrix (7), we obtainwhereAlso, for all integer , we haveBy multiplying (51) by from the right and (53) by from the left followed by subtraction and -integration from to , yieldsOn -integrating by parts givesIt was shown in Theorem 1 that as if and only if which is equivalent to as if and only if whereThis concludes thatif and only if . Hence, the proof is complete.

Lemma 4. Let be a square complex matrix satisfying the conditions (3) and (4). Then, we have

Proof. Since the function is even, we obtainFrom (13), we getThe interchange of the summation and integration is justified due to their convergence.
It has been shown in [23] that if , then has a unique square root with in the open right (complex) half plane. Now, replacing by , according to the former notice, we find thatIn view of the definition of -gamma function (16), the previous integral equals . Therefore,The Jacobi triple product can be written as [21].Replacing by with noting thatgivesSubstituting into (62) to obtain the desired result.

Theorem 3. Assume that and is a square complex matrix satisfying the conditions (3) and (4). Then, we have

Proof. The Rodrigues-type formula for the discrete -Hermite matrix polynomials was derived by the following equation [18]:whereTherefore, (47) givesIt is easy, from the definition of -derivative (8), to see thatUsing the identity with puttingto obtainOn -integrating by parts yieldsBy virtue of the results obtained in Theorem 1, the first term equals zero and by using the identity [18].we can arrive atIterating this process yieldsFrom the explicit formula (5) for the discrete -Hermite matrix polynomials at , we get and so from Lemma 4, we getSubstituting 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 and be a square complex matrix satisfying the conditions (3) and (4). Then, the discrete -Hermite matrix polynomial is an orthogonal polynomial with respect to the inner product defined in (47). Furthermore,where is the Kronecker delta.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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.