# Orthogonality Property of the Discreteq-Hermite Matrix Polynomials

Authors: Ahmed Salem; Faris Alzahrani; Moustafa El-Shahed
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we prove that the solution of the autonomousq-difference system DqYx=AYqx with the initial condition Y0=Y0 where A is a constant square complex matrix, Dq is the Jackson q-derivative and 0<q<1, is asymptotically stable if and only if ℜλ<0 for all λ∈σA where σA 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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## Body

## 1. Introduction

The beginning ofq-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 theq-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 Hnx,A;q by means of the generating matrix function(1)t;q∞−t;q∞xtA;q∞−1=∑n=0∞Hnx,A;qq;qntn,where A;q∞ is the q-shifted factorial matrix function defined [15] as follows:(2)A;q∞=limn⟶∞A;qn=limn⟶∞∏k=0n−1I−Aqk,and A is a square matrix satisfying the following conditions:(3)ℜz>0,for allz∈σA,where σA denotes the set of all eigenvalues of A, and(4)A<1andq−m∉σA,m∈ℕ0,for the operator norm ⋅ corresponding to the 2-norm. The discrete q-Hermite matrix polynomials has an explicit representation of the form.(5)Hnx,A;q=∑k=0n/2−1kqkk−1q;qnq2;q2kq;qn−2kxAn−2k,n∈ℕ0,where A is the square root of the matrix A in the sense of the matrix functional calculus A=exp1/2logA, and log denotes the principal branch of the complex logarithm, and satisfies the three terms recurrence relation.(6)Hnx,A;q=xAHn−1x,A;q−qn−21−qn−1Hn−2x,A;q,n∈ℕ,and q-difference matrix equation.(7)Dq2−q2−n1−qxADq+q2−n1−qnqAHnx,A;q=0,where Dqf is the q-derivative of a function f defined as follows:(8)Dqfx=fx−fxq1−qx,q≠1,x≠0.An easy consequence of the above definition, theq-derivative of two product functions is(9)Dqfxgx=gxDqfx+fqxDqgx=fxDqgx+gqxDqfx.The discreteq-Hermite matrix polynomials of type II hnx,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:(10)hnx,A;q=−inHnix,A;1q.The main goal of this paper is to investigate the orthogonality property of the discreteq-Hermite matrix polynomials of type I.


## 2. Asymptotically Stable Solution to Autonomousq-Difference System

The stability theory of differential and difference equations is of main interest in physical systems. The complete controllability and observability ofq-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.(11)DqYx=AYqx,Y0=Y0,where A is constant matrix in ℂn×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:(12)Yx=Y0∏n=0∞I+1−qxAqn=Y0EqxA,where EqA is the q-exponential matrix function and also can be defined as follows:(13)EqA=∑n=0∞qn2nq!An,where aq and nq! are the basic number and the basic factorial function defined, respectively, as follows:(14)aq=1−qa1−q,q≠1,and(15)nq!=nqn−1q⋯2q1q,n∈ℕwith0q!=1.Definition 1.
The solution of the autonomous system (11) is said to be stable if there exists ε>0 such that Yx<ε for all x>0. The solution is said to be asymptotically stable if, in addition to being stable, Yx⟶0 as x⟶∞.Definition 2.
Theq-gamma function is defined as follows:(16)Γqα=∫01/1−qtα−1Eq−qtdqt=∫0∞/1−qtα−1Eq−qtdqt,ℜα>0,and has the identity Γqα+1=αqΓqα, where the q-integration of Jackson in a generic interval a,b is given as follows:(17)∫abftdqt=∫0bftdqt−∫0aftdqt,where(18)∫0afxdqx=a1−q∑n=0∞faqnqn,provided the sum converges absolutely and the improper integral is defined in the following way (see [20, 21]):(19)∫0∞/1−qfxdqx=∑n=−∞∞qnfqn1−q.
The bilateralq-integral is defined as follows:(20)∫−∞∞fxdqx=1−q∑n=0∞qnfqn+f−qn.
Theq-integrating by parts is given for suitable functions f and g by(21)∫abfxDqgxdqx=fbgb−faga−∫abgqxDqfxdqx.
Let the two series(22)Sux=∑n=0∞1−qqn1+ux1−qqn,Tx=i∑k=0∞v1−qqk1+ux1−qqk1+λx1−qqk,be defined for large x and 0<q<1 where λ=u+iv∈ℂ and u,v∈ℝ. It is clear that(23)limx⟶∞TxSux=limx⟶∞∑k=0∞ivqk∑n=0∞qn1+ux1−qqk1+λx1−qqk/1+ux1−qqn=0,(24)limx⟶∞1xSλx=limx⟶∞1∑n=0∞x1−qqn/1+λx1−qqn=0.
The interchange of the limits and the series has verified due to the convergence of the series.Lemma 1.
For all complex numberλ, we have(25)limx⟶∞Eqλx=0,0<q<1,if and only if ℜλ<0.Proof.
From the definitions ofq-gamma function (16), we get(26)1=Γq1=−∫0∞/1−qDqEq−tdqt=1−limy⟶∞Eq−y1−q,which reveals that limy⟶∞Eq−y/1−q=0 or equivalently, limx⟶∞Equx=0 if u<0. Now, we can write(27)Eqλx=∏n=0∞1+λx1−qqn=∏n=0∞1+ux1−qqn∏n=0∞1+ivx1−qqn1+ux1−qqn=Equx∏n=0∞1+ivx1−qqn1+ux1−qqn,λ=u+iv.
Using L’Hospital rule gives(28)limx⟶∞Eqλx1+1ulimx⟶∞TxSux=0.
In view of (23), we get limx⟶∞Eqλx=0 if u=ℜλ<0. Now, let u≥0, then limx⟶∞Equx≠0 which reveals that limx⟶∞Eqλx≠0.Lemma 2.
For all complex numbersα and λ, we have(29)limx⟶∞xαEqλx=0,if and only if ℜλ<0 and ℜα≥0 or ℜλ≤0 and ℜα<0.Proof.
There are nine cases that can be rewritten in five casesCase 1.
ℜα≤0 and ℜλ<0,(30)limx⟶∞xαEqλx=limx⟶∞xαlimx⟶∞Eqλx=0.Case 2.
ℜα>0 and ℜλ<0, using L’Hospital rule gives(31)limx⟶∞xαEqλx1+αλlimx⟶∞1xSλx=0.
In view of (24), we get limx⟶∞xαEqλx=0.Case 3.
ℜα<0 and ℜλ=0, if ℑλ=0, then Eq0=1 and so(32)limx⟶∞xαEqλx=limx⟶∞xα=0.
Ifℑλ≠0, we find that(33)limx⟶∞xαEqλx≤limx⟶∞xℜα∏n=0∞1+vxqn,v=ℑλ.
The product above approaches infinity asx⟶∞ and so we can use the L’Hospital rule to obtain(34)limx⟶∞xℜα∏n=0∞1+vxqn1+ℜαvlimx⟶∞1xSvx=0,which reveals by using (24) that limx⟶∞xℜα∏n=0∞1+vxqn=0 and so limx⟶∞xαEqλx=0.Case 4.
ℜα≥0 and ℜλ≥0, then limx⟶∞Eqλx≠0 which yields that limx⟶∞xαEqλx≠0.Case 5.
ℜα<0 and ℜλ>0, let −k−1≤ℜα<−k for some positive integer k, then we have(35)limx⟶∞xαEqλx=limx⟶∞∑n=0kqn2λnnq!xα+n+limx⟶∞∑n=k+1∞qn2λnnq!xα+n.
It is obvious that the first limit equals zero and the last limit does not equal zero which means thatlimx⟶∞xαEqλx≠0.
These end the proof.Lemma 3.
For all complex numberλ, we have(36)limx⟶∞dndznEqxz|z=λ=0,n∈ℕ0,if and only if ℜλ<0.Proof.
The case ofn=0 is proven in the lemma above. When n=1, the logarithmic derivative for Eqxz with respect to z gives(37)ddzEqxz=xEqxzSxz.
Iterating this process yields(38)dndznEqxz=xnEqxzgxz,where gxz is a finite sum of the infinite series ∑k=0∞1−qqk/1+1−qxzqkrs and r,s∈1,2,…,n which tend to zero as x⟶∞. By virtue of the results obtained in the lemma above, we get(39)limx⟶∞dndznEqxz|z=λ=limx⟶∞xnEqλxlimx⟶∞gλx=0,ℜλ<0.
This ends the proof.Theorem 1.
The solution of the autonomousq-difference system (11) is asymptotically stable if and only if αA<0 for all x>0 where αA is the spectral abscissa defined as follows:(40)αA=maxℜz:z∈σA.Proof.
According to the Jordan matrix decomposition [22], with regard to the square matrix A∈ℂn×n, there exists an invertible matrix P such that(41)A=PJP−1=PdiagJ1,J2,…,JkP−1,where the Ji 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(42)Ji=λi1λi1⋱⋱λi1λini×ni,i=1,2,…k,where λi∈σA and ∑i=1kni=n. From the definition of q-exponential matrix function (13) and the relation (41), we get(43)EqxA=P∑m=0∞qm2diagJ1m,J2m,…,Jkmxmmq!P−1=PdiagEqxJ1,EqxJ2,…,EqxJkP−1,where the matrix EqxJi can be written in the form(44)EqxJi=fλf′λf″λ2…fnk−2λnk−2!fnk−1λnk−1!0fλf′λ⋯fnk−3λnk−3!fnk−2λnk−2!00fλ⋯fnk−4λnk−4!fnk−3λnk−3!⋮⋮⋮⋱⋮⋮000⋯fλf′λ000⋯0fλλ=λi,and fλ=Eqxλ. In view of the results obtained by Lemma 3, the limit of nonzero elements of EqxJi can be computed as follows:(45)limx⟶∞drdλrEqxλ|λ=λi=0,r=0,1,…nk−2,ℜλi<0,which concludes that(46)limx⟶∞‖EqxA‖=0,ℜλi<0,λi∈σA.
This completes the proof.


## 3. Orthogonality Property

Suppose that the inner productf,g for a suitable two matrix-valued functions f and g is defined as follows:(47)fx,gx=∫−∞∞wx,A;qfxgxdqx,where the weight function wx,A;q is defined as follows:(48)wx,A;q=qxA;q∞−qxA;q∞=q2x2A;q2∞.Definition 3.
LetPnx be a matrix polynomials for integer n≥0. We say that the sequence Pnxn≥0 is an orthogonal matrix polynomials sequence with respect to the inner product ⋅,⋅ provided for all nonnegative integers n and m.(1)
Pnx is a matrix polynomial of degree n with nonsingular leading coefficient.(2)
Pnx,Pmx=0 for all n≠m.(3)
Pnx,Pnx is invertible for n≥0.Theorem 2.
Assume thatn,m∈ℕ0 and A is a square complex matrix satisfying the conditions (3) and (4). Then, we have(49)Hnx,A;q,Hmx,A;q=0,n≠m.Proof.
According to theq-derivative of two product functions (9), we get(50)Dqwx,A;qDqHnx,A;q=wqx,A;qDq2Hnx,A;q+Dqwx,A;qDqHnx,A;q.
From theq-difference matrix (7), we obtain(51)Dqwx,A;qDqHnx,A;q=q2−nnqwqx,A;qAxDqHnx,A;q−Hnx,A;q1−q=q2−nnqwqx,A;qAHnx,A;q−Hnqx,A;q1−q−Hnx,A;q1−q=λnwqx,A;qHnqx,A;q,n∈ℕ0,where(52)λn=−q2−n1−qnqA.
Also, for all integerm∈ℕ0, we have(53)Dqwx,A;qDqHmx,A;q=λmwqx,A;qHmqx,A;q.
By multiplying (51) by Hmqx,A;q from the right and (53) by Hnqx,A;q from the left followed by subtraction and q-integration from −∞ to ∞, yields(54)λn−λmHnx,A;q,Hmx,A;q=λn−λmq∫−∞∞wqx,A;qHnqx,A;qHmqx,A;qdqx=q∫−∞∞Dqwx,A;qDqHnx,A;qHmqx,A;qdqx−q∫−∞∞Hnqx,A;qDqwx,A;qDqHmx,A;qdqx.
Onq-integrating by parts gives(55)λn−λmHnx,A;q,Hmx,A;q=qwx,A;qDqHnx,A;qHmx,A;q|−∞∞−qwx,A;qDqHmx,A;qHnx,A;q|−∞∞.
It was shown in Theorem1 that EqxA⟶0 as x⟶∞ if and only if αA<0 which is equivalent to Eq−xA=1−qxA;q∞⟶0 as x⟶∞ if and only if βA>0 where(56)βA=−α−A=minℜz:z∈σA.
This concludes that(57)limx⟶∞wx,A;q=limx⟶∞q2x2A;q2∞=0,if and only if βA>0. Hence, the proof is complete.Lemma 4.
LetA be a square complex matrix satisfying the conditions (3) and (4). Then, we have(58)∫−∞∞q2x2A;q2∞dqx=1−qq;q∞−q;q∞−1;q∞A−1.Proof.
Since the functionq2x2A;q2∞ is even, we obtain(59)∫−∞∞q2x2A;q2∞dqx=2∫0∞q2x2A;q2∞dqx=2∫0∞−qxA;q∞qxA;q∞dqx=2∫0∞EqqxA1−qEq−qxA1−qdqx.
From (13), we get(60)∫−∞∞q2x2A;q2∞dqx=2∑n=0∞qnn+1/2q;qnAn∫0∞xnEq−qxA1−qdqx.
The interchange of the summation and integration is justified due to their convergence.
It has been shown in [23] that if σA∩−∞,0=ϕ, then A has a unique square root with σA in the open right (complex) half plane. Now, replacing x by 1−qtA−1, according to the former notice, we find that(61)∫−∞∞q2x2A;q2∞dqx=21−qA−1∑n=0∞qnn+1/2nq!∫0∞/1−qtnEq−qtdqt.
In view of the definition ofq-gamma function (16), the previous integral equals Γqn+1=nq!. Therefore,(62)∫−∞∞q2x2A;q2∞dqx=21−qA−1∑n=0∞qnn+1/2.
The Jacobi triple product can be written as [21].(63)∑n=−∞∞−1nqn2xn=q;q∞x;q∞qx−1;q∞.
Replacingx by −q with noting that(64)∑n=−∞∞qnn+1/2=2∑n=0∞qnn+1/2,gives(65)∑n=0∞qnn+1/2=12q;q∞−q;q∞−1;q∞.
Substituting into (62) to obtain the desired result.Theorem 3.
Assume thatn∈ℕ0 and A is a square complex matrix satisfying the conditions (3) and (4). Then, we have(66)Hnx,A;q,Hnx,A;q=1−qq;qnq,−q,−1;q∞qn2A2n−1.Proof.
The Rodrigues-type formula for the discreteq-Hermite matrix polynomials was derived by the following equation [18]:(67)wx,A;qHnx,A;q=ξnDq−1nwx,A;q,n∈ℕ0,where(68)ξn=−1n1−qnqnn−3/2An.
Therefore, (47) gives(69)Hnx,A;q,Hnx,A;q=∫−∞∞wx,A;qHnx,A;qHnx,A;qdqx=ξn∫−∞∞Dq−1nwx,A;qHnx,A;qdqx.
It is easy, from the definition ofq-derivative (8), to see that(70)Dq−1fx=qDqfxq−1.
Using the identity with putting(71)fx=Dq−1n−1wx,A;q=ξ−1n−1wx,A;qHn−1x,A;q,to obtain(72)Hnx,A;q,Hnx,A;q=qξn∫−∞∞Dqfxq−1Hnx,A;qdqx.
Onq-integrating by parts yields(73)Hnx,A;q,Hnx,A;q=qξnfxq−1Hnx,A;q|−∞∞−∫−∞∞fxDqHnx,A;qdqx.
By virtue of the results obtained in Theorem1, the first term equals zero and by using the identity [18].(74)DqHnx,A;q=nqAHn−1x,A;q,we can arrive at(75)Hnx,A;q,Hnx,A;q=qn−11−qnAHn−1x,A;q,Hn−1x,A;q.
Iterating this process yields(76)Hnx,A;q,Hnx,A;q=qnn−1/2q;qnAnH0x,A;q,H0x,A;q.
From the explicit formula (5) for the discrete q-Hermite matrix polynomials at n=0, we get H0x,A;q=I and so from Lemma 4, we get(77)H0x,A;q,H0x,A;q=∫−∞∞q2x2A;q2∞dqx=1−qq;q∞−q;q∞−1;q∞A−1.
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 thatn∈ℕ0 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 ⋅,⋅ defined in (47). Furthermore,(78)Hnx,A;q,Hmx,A;q=1−qq;qnq,−q,−1;q∞qn2A2n−1δnm,where δnm is the Kronecker delta.



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