The q-Laguerre matrix polynomials

The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.

(1) I. An explicit expression for the Laguerre matrix polynomials, a three-term matrix recurrence relation, a Rodrigues formula and orthogonality properties are given in Jódar et al. (1994). The Laguerre matrix polynomials satisfy functional relations and properties which have been studied in Sastre (2001, 2004), Sastre and Jódar (2006a, b), Sastre and Defez (2006), . Recently, q-calculus has served as a bridge between mathematics and physics. Therefore, there is a significant increase of activity in the area of the q-calculus due to its applications in mathematics, statistics and physics. The one of the most important concepts in q-calculus is the Jackson q-derivative operator defined as which becomes the same as ordinary differentiation in the limit as q → 1. We shall use the q-analogue of the product rule Exton (1977) discussed a basic analogue of the generalized Laguerre equation by means of replacing the ordinary derivatives by the q-operator (3) and studied some properties of certain of its solutions. Moak (1981) introduced and studied the q-Laguerre polynomials for 0 < q < 1, where [a] q = (1 − q a )/(1 − q), (a; q) n is the q-shifted factorial defined as and [n] q ! is the q-factorial function defined as The q-Laguerre polynomials (5) appeared as a solution of the second order q-difference equation The q-Laguerre polynomials has been drawn the attention of many authors who proved many properties for it. For more details see Koekoek and Swarttouw (1998), Koekoek (1992).
As a first step to extend the matrix framework of quantum calculus, the q-gamma and q-beta matrix functions have been introduced and studied in Salem (2012). Also, the basic Gauss hypergeometric matrix function has been studied in Salem (2014).
In this paper, we extend the family of q-Laguerre polynomials (5) of complex variables to q-Laguerre matrix polynomials by means of studying the solutions of the second order matrix q-difference equations where , α ∈ C and A, C and Y(x) are square matrices in C r×r . The orthogonality property, explicit formula, Rodrigues-type formula, three-terms recurrence relations, generating functions and other properties will be derived.
For the sake of clarity in the presentation, we recall some properties and notations, which will be used below. Let ||A|| denote the norm of the matrix A, then the operator norm corresponding to the two-norm for vectors is where σ (A) is the spectrum of A: the set of all eigenvalues of A and A * denotes the transpose conjugate of A. If f(z) and g(z) are holomorphic functions of the complex variable z, which are defined in an open set of the complex plane and if A is a matrix in C N ×N such that σ (A) ⊂ �, then from the properties of the matrix functional calculus (Dunford and Schwartz 1956), it follows that f The logarithmic norm of a matrix A is defined as (Sastre and Defez 2006) Suppose the number μ(A) such that By Higueras and Garcia-Celaeta (1999), it follows that �e At � ≤ e tµ(A) for t ≥ 0, we have If A(k, n) and B(k, n) are matrices on C N ×N for n, k ∈ N 0 , it follows that (Defez and Jódar 1998)

Matrix q-difference equation
The following lemmas will be used in this section.
Lemma 1 Let α and a be complex number with R(a) > 0, then we have where e q (x) is the q-analogue of the exponential function defined as Proof Let f (x) = x α e q (−ax). Taking the limit of logarithm of the function f(x) as x → ∞ (11), gives Taking n ∈ N 0 such that n ≥ α yields This means that lim x→∞ f (x) = e −∞ = 0 which completes the proof.
Lemma 2 Suppose that A ∈ C r×r satisfying the condition and let be a complex number with R( ) > 0. Then it follows that and where P n (x) is a matrix polynomials of degree n ∈ N 0 .
Proof From (7), we get Since e q (−qx )P n (x) is bounded as x → 0, it follows that (13) holds. From (7), we get Salem SpringerPlus (2016) 5:550 Let P n (x) = a 1 x n + a 2 x n−1 + · · · + a n and let 0 ≤ k ≤ n, then Lemma 1 gives it follows that which ends the proof.
Theorem 3 Let m, n ∈ N 0 , A ∈ C r×r satisfying the condition (12) and C ∈ C r×r is invertible and depends on A. Let Y m and Y n are solutions of matrix q-difference equation (6) corresponding to α m and α n respectively, then we get where the q-integral is the inverse of q-derivative (3) defined as Proof By virtue of (4), the matrix q-difference equation (6) can be read as Since Y m and Y n are solutions of (16) corresponding to α m and α n respectively, then we can easily obtain On q-integrating both sides from 0 to ∞ and by Lemma 2 and hypothesis α m � = α n yields which ends the proof. Now, let us suppose that the solution of (6) has the form where 0 r×r is the null matrix in C r×r .
To determine the matrices a k . Taking formal q-derivatives (3) of Y (x), it follows that Substituting into (6) would yield Equating the coefficients of x k , k ∈ N 0 would yield and which can be read as For existence of the second order q-difference equation, we seek the sufficient condition We have to suppose that q −k � ∈ σ (q A ), k ∈ N 0 to ensure that the relevant (I − q A+kI ) exists. Therefore, (18) gives which leads to Letting the boundary condition Y (0) = (q A+I ;q) n (q;q) n which reveals that a 0 = (q A+I ;q) n (q;q) n and so we can seek the following definition for the q-Laguerre matrix function which verified the Eq. (6) Definition 4 Let n ∈ N 0 , be a complex number with R( ) > 0 and A ∈ C r×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all 0 ≤ k ≤ n. The q-Laguerre matrix polynomials can be defined as Remark 5 When letting q → 1, the matrix q-difference equation (6) tends to the matrix differential equation (1) and also the q-Laguerre matrix polynomials (19) approache to the Laguerre matrix polynomials (2). We proved that the q-Laguerre matrix polynomials (19) hold for μ(A) > −1 but Jódar et al. (1994) proved that the Laguerre matrix poly- It is worth noting that the important relation between β(A) and μ(A), which states β(A) >μ(A) (Salem 2012). Therefore, the definition of the Laguerre matrix polynomials (2) can be extended for μ(A) > −1.

Generating functions
The basic hypergeometric series is defined as Gasper and Rahman (2004) for all complex variable z if r ≤ s, 0 < |q| < 1 and for |z| < 1 if r = s + 1, where Notice that the q-shifted function (a; q) n has the summation Koekoek and Swarttouw (1998) where n k q is the q-binomial coefficients defined as Also it has the well-known identities The q-shifted factorial matrix function was defined in Salem (2012) as and satisfies Furthermore, if A < 1 and q −k � ∈ σ (A), k ∈ N 0 , the infinite product (26) converges invertibly and In Salem (2012) a proof of the matrix q-binomial theorem can be found for all commutative matrices A, B ∈ C r×r and q −k � ∈ σ (A), k ∈ N 0 .
Lemma 7 Let A and C are matrices inC r×r such that q −n � ∈ σ (C) for all n ∈ N 0 and a ∈ C, the matrix functions and converge absolutely.
Proof The condition q −n � ∈ σ (C) guarantees that I − Cq n is invertible for all integer n ≥ 0. Now take n large enough so that �C� < |q| −n , by the perturbation lemma (Constantine and Muirhead 1972) one gets If we take and by the relation (35), we get Using the ratio test and the perturbation lemma (34), one finds Thus, the matrix power series (32) is absolutely convergent. Similarly, (33) can be proved. This ends the proof. Since the function 1 φ 1 (a; 0; q, z) is analytic for all complex numbers a and z, the matrix functional calculus tells that the matrix function 1 φ 1 (a; 0; q, A) is also convergent for all complex number a and for all matrices A ∈ C r×r . Lemma 8 Let A be matrix inC r×r such that q −n � ∈ σ (C) for all n ∈ N 0 and a be a complex number. We have the transformation Proof The relation (21) can be exploited to prove the transformation Using the relations (9) and (26)   q n(n−1) (C; q) −1 n (q; q) n A n ≤ ∞ n=0 q n(n−1) F (n) (q; q) n �A� n .
Corollary 10 Let n ∈ N 0 , be a complex number with R( ) > 0 and A ∈ C r×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all 0 ≤ k ≤ n. The q-Laguerre matrix polynomials can be defined as Proof The generating function (37) can be expanded as This ends the proof.

Remark 11
In view of the explicit expressions of the q-Laguerre matrix polynomials (19) and (41), with replacing q A+I and − x(1 − q) by A and x, respectively, we can derive the matrix transformation which tends to the transformation (36) as n → ∞.
n (x; q)t n .

Recurrence relations and Rodrigues-type formula
This section is devoted to introduce some recurrence relations and Rodrigues type formula for the q-Laguerre matrix polynomials.

Theorem 12
Let be a complex number with R( ) > 0 and A ∈ C r×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all 0 ≤ k ≤ n. Then, the q-Laguerre matrix polynomials satisfy the three-term matrix recurrence relation Proof Let the matrix-valued function Using Jackson q-derivatives operator (3) and the q-analogue of the product rule (4) give Inserting the above relation into the generating function (37) with taking the q-derivative of the right hand side yields Equating to the zero matrix the coefficient of each power t n it follows that and (43) n (x; q).
Therefore the q-Laguerre matrix polynomials satisfy the three-term matrix recurrence relation (43).

Theorem 13
Let be a complex number with R( ) > 0 and A ∈ C r×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all 0 ≤ k ≤ n. Then, the q-Laguerre matrix polynomials satisfy the matrix relation Proof It is not difficult, by using (19) to see that the q-Laguerre matrix polynomials satisfy the forward shift operator which is equivalent to By iteration this process k-times, we can get the relation When k = n, we obtain which can be also obtained from nth term of (19) with fact that D n q x n = [n] q !. It is easy to show that From (37), we can deduce that which gives In view of iteration the above formula, we get the desired result.
In order to obtain the Rodrigues-type formula for the q-Laguerre matrix polynomials, we derive the following theorem.
n−k (xq k ; q), k = 0, 1, . . . , n; n ∈ N 0 . Theorem 14 (Rodrigues-type formula) Let be a complex number with R( ) > 0 and A ∈ C r×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all 0 ≤ k ≤ n. Then, the Rodrigues-type formula for the q-Laguerre matrix polynomials can be provided as Proof Using (3) yields and which can be rewrite by using (31) as From the Leibniz's rule for the nth q-derivative of a product rule Koekoek and Swarttouw (1998) and the properties of the matrix functional calculus, it follows that Inserting the last side of relation (22) to obtain the Rodrigues-type formula for the q-Laguerre matrix polynomials.

Orthogonality property
Suppose that the inner product f , g for a suitable two matrix-valued functions f and g is defined as Let P n (x) be a matrix polynomials for n ≥ 0. We say that the sequence {P n (x)} n≥0 is an orthogonal matrix polynomials sequence with respect to the inner product , provided for all nonnegative integers n and m n k q D n−k q f xq k D k q g (x), n ∈ N 0 D n q x A+nI e q (− x) = x A e q (− x) n k=0 (−1) k n k q q A+I ; q −1 k q A+I ; q n (1 − q) n−k q kA+k 2 I k x k .
On q-integrating by parts, which states we find which can be rewritten by Rodrigues-type formula (46) as By using Lemma 2, we obtain Using the q-analogue of the integration theorem by change of variable from qx to x yields which leads to Hence, from (52), we have the desired results.
Theorem 16 Let us assume that A ∈ C r×r satisfying the condition μ(A) > 0, then we have where Ŵ q (A) is the q-gamma matrix function defined by Salem (2012) as x A+nI e q (− qx)L (A+I, ) n−1 (qx; q)d q x. (53)  Using (29) followed by (23) when n = k, we can deduce that which concludes that In view of (56)-(58), we obtain An important relation for the q-gamma matrix function was obtained by Salem (2012) as which reveals that This completes the proof. The results proved in this section can be summarized in the following theorem: Theorem 17 Let n ∈ N 0 , be a complex number with R( ) > 0 and A ∈ C r×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all k ∈ N 0 , then the q-Laguerre matrix polynomials sequence {L (A, ) n (x; q)} n≥0 is an orthogonal matrix polynomials sequence with respect to the inner product

Conclusion
In our work, we introduce the q-Laguerre matrix polynomials (19) hold for μ(A) > −1 which verifies the second-order matrix difference equation (6). Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.