Salem SpringerPlus (2016)5:550 DOI 10.1186/s40064-016-2178-5
Open Access
RESEARCH
The q‑Laguerre matrix polynomials Ahmed Salem* *Correspondence: [email protected] Department of Basic Science, Faculty of Information Systems and Computer Science, October 6 University, Sixth of October City, Egypt
Abstract 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. Keywords: q-Laguerre matrix polynomials, q-Gamma matrix function, Matrix functional calculus, Three terms recurrence relation, Rodrigues-type formula Mathematics Subject Classification: 33D05, 33D45, 15A16
Background The study of functions of matrices is a very popular topic in the Matrix Analysis literature. Some basic references are Gantmacher (1998), Higham (2008) and Horn and Johnson (1991). The subject of the orthogonal polynomials cuts across a large piece of mathematics and its applications. Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949. In the last two decades this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of n (Duran and Grunbaum 2005). The second-order matrix differential equations of the form xY ′′ (x) + (A + I − xI)Y ′ (x) + nY (x) = 0,
n ∈ N0 , A, Y (x) ∈ Cr×r
(1)
have been introduced and investigated in Jódar et al. (1994), where Cr×r denotes the vector space containing all square matrices with r rows and r columns with entries in the complex number C, ∈ C and x is a real number. The explicit expression for the nth (A, ) Laguerre matrix polynomial Ln (x) which is a solution of the second order matrix differential equation (1), has the form
L(A, ) (x) = n
n (−1)k k k (A + I)n (A + I)−1 k x , k!(n − k)! k=0
R( ) > 0
(2)
© 2016 Salem. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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where (A + I)n = (A + I)(A + 2I) . . . (A + (n − 1)I), n ∈ N and (A + I)0 = 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 Jódar and Sastre (2001, 2004), Sastre and Jódar (2006a, b), Sastre and Defez (2006), Sastre et al. (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
(Dq f )(x) =
f (x) − f (qx) , (1 − q)x
(3)
q �= 1, x �= 0
which becomes the same as ordinary differentiation in the limit as q → 1. We shall use the q-analogue of the product rule
(4)
(Dq fg)(x) = f (x)(Dq g)(x) + (Dq f )(x)g(qx).
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
L(α) n (x; q)
=
q α+1 ; q
n
(q; q)n
k n −n k(k−1) q ; q k q 2 q n+α+1 x , [k]q ! q α+1 ; q k k=0
α > −1, n ∈ N0
(5)
for 0 < q < 1, where [a]q = (1 − q a )/(1 − q), (a; q)n is the q-shifted factorial defined as
(a; q)n =
n−1 k=0
(1 − aq k ),
n∈N
with (a; q)0 = 1
and [n]q ! is the q-factorial function defined as
[n]q ! = [n]q [n − 1]q . . . [1]q ,
n∈N
with [0]q ! = 1.
The q-Laguerre polynomials (5) appeared as a solution of the second order q-difference equation
xDq2 y(x) + [α + 1]q − q α+2 x (Dq y)(qx) + [n]q q α+1 y(qx) = 0,
α > −1.
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).
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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
x(Dq2 Y )(x) + [A + I]q − x q A+2I (Dq Y )(qx) + [α]q CY (qx) = 0
(6)
where , α ∈ C and A, C and Y(x) are square matrices in Cr×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
||A|| = max x� =0
√ ||Ax||2 : ∈ σ (A∗ A) = max ||x||2
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 CN ×N such that σ (A) ⊂ �, then from the properties of the matrix functional calculus (Dunford and Schwartz 1956), it follows that f (A)g(A) = g(A)f (A). The logarithmic norm of a matrix A is defined as (Sastre and Defez 2006)
�I + hA� − 1 = max z : z ∈ σ A + A∗ /2 . h h→0
µ(A) = lim
˜ such that Suppose the number µ(A) µ(A) ˜ = −µ(−A) = min z : z ∈ σ A + A∗ /2 .
By Higueras and Garcia-Celaeta (1999), it follows that �eAt � ≤ etµ(A) for t ≥ 0, we have µ(A) t , if t ≥ 1, A �t � ≤ (7) ˜ , if 0 ≤ t ≤ 1. t µ(A)
If A(k, n) and B(k, n) are matrices on CN ×N for n, k ∈ N0, it follows that (Defez and Jódar 1998) ∞ ∞
B(k, n) =
∞ n
B(k, n − k)
(8)
∞ n
B(k, n) =
∞ ∞
B(k, n + k).
(9)
n=0 k=0
n=0 k=0
and
n=0 k=0
n=0 k=0
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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
lim xα eq (−ax) = 0
(10)
x→∞
where eq (x) is the q-analogue of the exponential function defined as
eq (x) =
∞ xk 1 = , [k]q ! (x(1 − q); q)∞ k=0
|x| < (1 − q)−1 .
(11)
Proof Let f (x) = xα eq (−ax). Taking the limit of logarithm of the function f(x) as x → ∞ (11), gives
lim log f (x) = lim
x→∞
x→∞
α log x −
∞ k=0
log 1 + axq k
Taking n ∈ N0 such that n ≥ α yields
lim log f (x) = lim
x→∞
x→∞
log
xα n−1 k=0
1 + axq k
−
∞ k=n
log 1 + axq k = −∞
This means that limx→∞ f (x) = e−∞ = 0 which completes the proof. Lemma 2 Suppose that A ∈ Cr×r satisfying the condition
(12)
µ(A) ˜ > −1 and let be a complex number with R( ) > 0. Then it follows that
lim xA+I eq (−qx )Pn (x) = 0
(13)
lim xA+I eq (−qx )Pn (x) = 0
(14)
x→0
and x→∞
where Pn (x) is a matrix polynomials of degree n ∈ N0. Proof From (7), we get ˜ �xA+I � ≤ xµ(A)+1 ,
x→0
Since eq (−qx )Pn (x) is bounded as x → 0, it follows that (13) holds. From (7), we get
�xA+I � ≤ xµ(A)+1 ,
x→∞
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Let Pn (x) = a1 xn + a2 xn−1 + · · · + an and let 0 ≤ k ≤ n, then Lemma 1 gives
lim xµ(A)+k+1 eq (−qx ) = 0,
R( ) > 0
x→∞
it follows that
lim xA eq (−qx )Pn (x) = 0,
R( ) > 0
x→∞
which ends the proof. Theorem 3 Let m, n ∈ N0, A ∈ Cr×r satisfying the condition (12) and C ∈ Cr×r is invertible and depends on A. Let Ym and Yn are solutions of matrix q-difference equation (6) corresponding to αm and αn respectively, then we get ∞ xA eq (−qx )Ym (qx)Yn (qx)dq x = 0, m �= n (15) 0
where the q-integral is the inverse of q-derivative (3) defined as
0
∞
f (x)dq x = (1 − q)
∞
k=−∞
qk f qk
Proof By virtue of (4), the matrix q-difference equation (6) can be read as
Dq xA+I eq (−qx )(Dq Y )(x) (x) + [α]q xA Ceq (−qx )Y (qx) = 0
(16)
Since Ym and Yn are solutions of (16) corresponding to αm and αn respectively, then we can easily obtain
[αm ]q − [αn ]q xA Ceq (−qx )Ym (qx)Yn (qx) = Ym (qx) Dq xA+I eq (−qx )(Dq Yn )(x) (x) − Dq xA+I eq (−qx )(Dq Ym )(x) (x)Yn (qx) = Dq xA+I eq (−qx )(Dq Yn )(x)Ym (x) − xA+I eq (−qx )Yn (x)(Dq Ym )(x) (x)
On q-integrating both sides from 0 to ∞ and by Lemma 2 and hypothesis αm � = αn yields ∞ ([am ]q − [an ]q )C xA eq (−qx )Ym (qx)Yn (qx)dq x = 0, m �= n 0
which ends the proof. Now, let us suppose that the solution of (6) has the form
Y (x) =
∞ k=0
ak xk ,
ak ∈ Cr×r ,
where 0r×r is the null matrix in Cr×r.
a0 �= 0r×r
(17)
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To determine the matrices ak . Taking formal q-derivatives (3) of Y (x), it follows that
(Dq Y )(x) =
∞ k=0
[k + 1]q ak+1 xk ,
and
∞ Dq2 Y (x) = [k + 1]q [k + 2]q ak+2 xk k=0
Substituting into (6) would yield ∞ k=0
∞ [k + 1]q [k + 2]q ak+2 xk+1 + [A + I]q − x q A+2I [k + 1]q ak q k xk k=0
+ [α]q C
∞ k=0
ak q k xk = 0
Equating the coefficients of xk , k ∈ N0 would yield
[A + I]q a1 + [α]q Ca0 = 0 and
[k]q + [A + I]q q k [k + 1]q ak+1 − q k [k]q q A+I − [α]q C ak = 0,
k∈N
which can be read as
[A + kI]q [k]q ak − q k−1 [k − 1]q q A+I − [α]q C ak−1 = 0,
(18)
k∈N
For existence of the second order q-difference equation, we seek the sufficient condition
C = q A+I ,
and α = n
for some non-negative integer n
We have to suppose that q −k � ∈ σ (q A ), k ∈ N0 to ensure that the relevant (I − q A+kI ) exists. Therefore, (18) gives
ak =
[k − n − 1]q A+(n+1)I [A + kI]−1 ak−1 , q q [k]q
k∈N
which leads to
ak =
q −n ; q
k
k q 2 k
[k]q !
q A+I ; q
−1 k
q (A+(n+1)I)k a0 A+I
A+I
n n which reveals that a0 = (q(q;q);q) and Letting the boundary condition Y (0) = (q(q;q);q) n n so we can seek the following definition for the q-Laguerre matrix function which verified the Eq. (6)
Definition 4 Let n ∈ N0, be a complex number with R( ) > 0 and A ∈ Cr×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
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L(A, ) (x; q) = n
n −n q ; q kq
k 2 k x k
[k]q !(q; q)n
k=0
q A+I ; q
−1 k
q A+I ; q
n
q (A+(n+1)I)k .
(19)
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 ˜ > −1 but Jódar et al. (1994) proved that the Laguerre matrix poly(19) hold for µ(A) nomials (2) hold for R(z) > −1 for all z ∈ σ (A) which equivalently β(A) > −1 where β = min{R(z) : z ∈ σ (A)}. It is worth noting that the important relation between β(A) ˜ ˜ , which states β(A) > µ(A) (Salem 2012). Therefore, the definition of the and µ(A) ˜ > −1. Laguerre matrix polynomials (2) can be extended for µ(A)
Generating functions The basic hypergeometric series is defined as Gasper and Rahman (2004)
a1 ,a2 ,...,ar r φs b1 ,b2 ,...,bs ; q, z
= r φs (a1 , a2 , . . . , ar ; b1 , b2 , . . . , bs ; q, z) =
∞ n s−r+1 (a1 , a2 , . . . , ar ; q)n zn, (−1)n q 2 (q, b1 , b2 , . . . , bs ; q)n
(20)
n=0
for all complex variable z if r ≤ s, 0 < |q| < 1 and for |z| < 1 if r = s + 1, where
(a1 , a2 , . . . , ar ; q)n = (a1 ; q)n (a2 ; q)n . . . (ar ; q)n Notice that the q-shifted function (a; q)n has the summation Koekoek and Swarttouw (1998)
(a; q)n = where
n
k q
n n k=0
k
q
k q 2 (−a)k
(21)
is the q-binomial coefficients defined as
−n n q ;q k n (q; q)n = (−1)k q kn− 2 . = (q; q)k (q; q)n−k (q; q)k k q
(22)
Also it has the well-known identities
(a; q)n−k
q k (a; q)n = 1−n −1 q − (q a ; q)k a
(a; q)n =
(a; q)∞ , (aq n ; q)∞
k −nk 2 ,
a �= 0,
k = 0, 1, 2, . . . n
(23)
and
n ∈ N0 .
(24)
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The q-shifted factorial matrix function was defined in Salem (2012) as
(A; q)0 = I,
(A; q)n =
n−1 k=0
I − Aq k ,
(25)
n∈N
and satisfies
(A; q)∞ = lim
n→∞
n−1 k=0
I − Aq k =
∞
I − Aq k =
k=0
∞ (−1)k q k=0
k 2
(q; q)k
Ak .
(26)
Furthermore, if A < 1 and q −k � ∈ σ (A), k ∈ N0, the infinite product (26) converges invertibly and
(A; q)−1 ∞ =
∞ k=0
Ak . (q; q)k
(27)
In Salem (2012) a proof of the matrix q-binomial theorem can be found ∞ (B; q)k k=0
(q; q)k
Ak = (A; q)−1 ∞ (AB; q)∞ ,
(28)
for all commutative matrices A, B ∈ Cr×r and q −k � ∈ σ (A), k ∈ N0. For complete this section, we need the following: Lemma 6 Let A be a square matrix inCr×r and n ∈ N0. Then, we have the following three identities ••
••
n
(A; q)n = (A; q)∞ Aq ; q
−1 ∞
=
n n k=0
k
q
k q 2 (−A)k ,
�A� < 1
(29)
where q −m
(A; q)−1 n
�∈ σ (A), m = n, n + 1, . . . , and its reciprocal ∞ n (q n ; q)k k = = (A; q)−1 Aq ; q A , �A� < 1 ∞ ∞ (q; q)k k=0
where q −k
•• Aq k ; q
(30)
�∈ σ (A), k ∈ N0. Also the identity
n−k
= (A; q)−1 k (A; q)n ,
k = 0, 1, . . . , n
holds for q −r � ∈ σ (A), r = 0, 1, . . . , k.
(31)
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Proof Let A < 1 and q −m � ∈ σ (A), m = n, n + 1, . . ., then we have
(A; q)n = =
n−1 k=0 ∞ k=0
1 − Aq k
1 − Aq k
= (A; q)∞
∞
k=0
∞
1 − Aq k
−1
k=n
1 − Aq k
∞ k=n
1 − Aq k+n
−1
1 − Aq k
−1
−1 = (A; q)∞ Aq n ; q ∞
Using (28) and (23) yields (29). The relation (30) comes immediately from (29) and (28). Also (31) can be easily obtained. Lemma 7 Let A and C are matrices inCr×r such that q −n � ∈ σ (C) for all n ∈ N0 and a ∈ C, the matrix functions 0 φ1 (−; C; q, A)
=
∞ q n(n−1) n=0
(q; q)n
n (C; q)−1 n A
(32)
and 0 φ2 (−; C, a; q, A)
=
3n(n−1) ∞ (−1)n q 2 n (C; q)−1 n A (q; q)n (a; q)n
(33)
n=0
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)
�A−1 − B−1 � ≤
�B−1 �2 �A − B� , 1 − �B−1 ��A − B�
(34)
one gets
−1 −1 � I − qn C � = � I − qn C − I + I� −1 − I� + 1 ≤ � I − qn C q n �C� +1 ≤ 1 − q n �C� ln �C� 1 . , n>− = 1 − q n �C� ln |q|
If we take
F (n) =
1, −1 n−1 k �, k=0 � I − q C
n = 0, n ≥ 1,
(35)
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and by the relation (35), we get
∞ n(n−1) ∞ (C; q)−1 q n(n−1) F (n) q n �0 φ1 (−; C; q, A)� ≤ An ≤ �A�n . (q; q)n (q; q)n n=0
n=0
Using the ratio test and the perturbation lemma (34), one finds
q n(n+1) F (n + 1)(q; q) �A�n+1 n lim n→∞ q n(n−1) F (n)(q; q)n+1 �A�n �(I − q n C)−1 � 2n q �A� n→∞ 1 − q n+1
= lim
≤ lim n→∞
q 2n �A� = 0. 1 − q n+1 (1 − q n �C�)
Thus, the matrix power series (32) is absolutely convergent. Similarly, (33) can be proved. This ends the proof. Since the function 1 φ1 (a; 0; q, z) is analytic for all complex numbers a and z, the matrix functional calculus tells that the matrix function 1 φ1 (a; 0; q, A) is also convergent for all complex number a and for all matrices A ∈ Cr×r. Lemma 8 Let A be matrix inCr×r such that q −n � ∈ σ (C) for all n ∈ N0 and a be a complex number. We have the transformation 1 φ1 (a; 0; q, A)
(36)
= (A; q)∞0 φ1 (−; A; q, aA)
Proof The relation (21) can be exploited to prove the transformation 1 φ1 (a; 0; q, A)
=
∞ (−1)n q
n 2 (a; q)n
(q; q)n
n=0
n
A
=
∞ (−1)n q n=0
n 2
(q; q)n
n
A
n n k=0
k
q
k q 2 (−a)k
Using the relations (9) and (26) lead to n+k k
+ ∞ ∞ 2 n + k (−1)n+k q 2 1 φ1 (a; 0; q, A) = (q; q)n+k k q
(−a)k
An+k
n=0 k=0
= =
∞ q k(k−1) ak k=0 ∞ k=0
(q; q)k
q k(k−1) ak (q; q)k
k
A
∞ (−1)n q n=0
n 2 +nk
(q; q)n
An
Ak (Aq k ; q)∞
Inserting the relation (30) into the above relation gives 1 φ1 (a; 0; q, A)
= (A; q)∞
This ends the proof.
∞ q k(k−1) ak k=0
(q; q)k
Ak (A; q)−1 k = (A; q)∞0 φ1 (−; A; q, aA)
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Theorem 9 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all k ∈ N0. The q-Laguerre matrix polynomials have the generating functions ∞
1 A+I t = L(A, ) (x; q)t n , 1 φ1 − x(1 − q); 0; q, q n (t; q)∞
(37)
n=0
∞ −1 (t; q)∞ tq −A−I ; q q −n(A+I) L(A, ) (x)t n 0 φ1 (−; t; q, − tx(1 − q)) = n ∞
(38)
n=0
∞
−1 1 A+I ; q, −q A+I (1 − q) xt = q A+I ; q L(A, ) (x; q)t n 0 φ1 −; q n n (t; q)∞
(39)
n=0
for all |t| < 1 and
(t; q)∞0 φ2 −; q A+I , t; q, − tx(1 − q)q A+I
=
∞ n=0
n −1 n q A+I ; q L(A, ) (x)t n (40) (−1) q 2 n n
for all t ∈ C. Proof The left hand side of (37) can be rewritten by means of using the transformation (36) as follows
1 A+I t 1 φ1 − x(1 − q); 0; q, q (t; q)∞ 1 A+I = ; q 0 φ1 −; tq A+I ; q, − xt(1 − q)q A+I tq ∞ (t; q)∞ ∞ (−1)n q n(n−) n xn t n A+I −1 n(A+I) 1 A+I tq ;q tq ;q q , = n ∞ (t; q)∞ [n]q !
LHS =
n=0
|t| < 1
Using (30) followed by (28) give
∞ (−1)n q n(n−1) n xn t n tq A+(n+1)I ; q ∞ n(A+I) LHS = q [n]q ! (t; q)∞ n=0 ∞ ∞ (−1)n q n(n−1) n xn t n q A+(n+1)I ; q k k n(A+I) t q = [n]q ! (q; q)k n=0
k=0
In view of (8), we get
LHS =
∞ n=0
t
n
n (−1)k q k(k−1) k xk k=0
[k]q !(q; q)n−k
q A+(k+1)I ; q
n−k
q k(A+I)
Inserting (23) and (31) with taking into account the definition of q-Laguerre matrix polynomials (19) to obtain the right hand side of (37). Similarly, we can find (38) with
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−A−I � < 1 which is equivalent to noting that the convergence of (tq −A−I ; q)−1 ∞ needs �tq A+I µ(A)+1 ˜ |t| < �q �≤q < 1. In order to prove (39), we have
1 A+I ; q, −q A+I (1 − q) xt 0 φ1 −; q (t; q)∞ ∞ ∞ t n (−1)n q n(n−1) n xn t n A+I −1 (A+I)n q ;q q = n (q; q)n [n]q ! =
n=0 ∞
n=0
tn
n=0
n k=0
(−1)k q k(k−1) k xk A+I −1 (A+I)k q q ;q k [k]q !(q; q)n−k
Using the well-known identity (23) to obtain (39). (40) is similar to (37) and (38). Corollary 10 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×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
L(A, ) (x; q) = n
n (q −n ; q)k (− x(1 − q); q)k
(q; q)k
k=0
q (A+(n+1)I)k .
(41)
Proof The generating function (37) can be expanded as
1 A+I t 1 φ1 − x(1 − q); 0; q, q (t; q)∞ n ∞ ∞ n n t (−1) q 2 (− x(1 − q); q)n (A+I)n n = q t (q; q)n (q; q)n n=0
n=0
k n ∞ k (−1) q 2 (− x(1 − q); q)k (A+I)k = q tn (q; q)k (q; q)n−k
= =
n=0
n=0
∞
n (q −n ; q)k (− x(1 − q); q)k
n=0
∞
tn
n=0
(q; q)k (q; q)n
q (A+(n+1)I)k
L(A, ) (x; q)t n . n
n=0
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 −n n = (A; q)n1 φ1 q −n ; A; q, xAq n (42) 2 φ1 q , x; 0; q, Aq which tends to the transformation (36) as n → ∞.
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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 ∈ Cr×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 (A, ) [n + 1]q Ln+1 (x; q) + xq A+(2n+1)I − q[n]q − [A + (n + 1)I]q L(A, ) (x; q) n (43) (A, ) + [A + nI]q Ln−1 (x; q) = 0, n ∈ N. Proof Let the matrix-valued function
F (x, t, A) =
1 A+I t 1 φ1 − x(1 − q); 0; q, q (t; q)∞
(44)
Using Jackson q-derivatives operator (3) and the q-analogue of the product rule (4) give
(Dq F )(t) =
1 A+I t 1 φ1 − x(1 − q); 0; q, q (1 − q)(t; q)∞ q A+I A+I − qt 1 φ1 − x(1 − q); 0; q, q (1 − q)(qt; q)∞ + x(1 − q)1 φ1 − x(1 − q); 0; q, q A+I q 2 t
Inserting the above relation into the generating function (37) with taking the q-derivative of the right hand side yields ∞ n=0
∞ n A+(n+1)I (x; q)t − q (x; q)t n+1 1 − q A+(n+1)I L(A, ) 1 − q L(A, ) n n n=0
− x(1 − q)q A+I =
∞ n=0
∞
L(A, ) (x; q)q 2n t n n
n=0
(A, )
(1 − q n+1 )Ln+1 (x; q)t n − q
∞ n=0
(A, )
(1 − q n+1 )Ln+1 (x; q)t n+1
Equating to the zero matrix the coefficient of each power t n it follows that (A, )
L1
(A, ) (x; q) = [A + I]q − xq A+I L0 (x; q)
and
(A, ) 1 − q A+(n+1)I L(A, ) (x; q) − 1 − q A+nI qLn−1 (x; q) n
− x(1 − q)q A+(2n+1)I L(A, ) (x; q) n (A, ) = 1 − q n+1 Ln+1 (x; q) − 1 − q n qL(A, ) (x; q). n
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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 ∈ Cr×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 n
(A, )
q i Li
i=0
(x; q) = Ln(A+I, ) (x; q),
n ∈ N0 .
(45)
Proof It is not difficult, by using (19) to see that the q-Laguerre matrix polynomials satisfy the forward shift operator (A+I, )
L(A, ) (x; q) − L(A, ) (qx; q) = − x(1 − q)q A+I Ln−1 n n
(qx; q)
which is equivalent to (A+I, )
Dq L(A, ) (x; q) = − q A+I Ln−1 n
(qx; q).
By iteration this process k-times, we can get the relation
(xq k ; q), k = 0, 1, . . . , n; Dqk L(A, ) (x; q) = (−1)k k q k(A+kI) L(A+kI, ) n n−k
n ∈ N0 .
When k = n, we obtain
Dqn L(A, ) (x; q) = (−1)n n q n(A+nI) , n
n ∈ N0
which can be also obtained from nth term of (19) with fact that Dqn xn = [n]q !. It is easy to show that
(1 − t)F (x, t, A + I) = F (x, qt, A) From (37), we can deduce that ∞ n=0
Ln(A+I, ) (x; q)t n −
∞ n=0
Ln(A+I, ) (x; q)t n+1 =
∞
q n L(A, ) (x; q)t n n
n=0
which gives (A+I, )
q n L(A, ) (x; q) = L(A+I, ) (x; q) − Ln−1 n n
(x; q).
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.
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Theorem 14 (Rodrigues-type formula) Let be a complex number with R( ) > 0 and A ∈ Cr×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
[n]q !xA eq (− x)L(A, ) (x; q) = Dqn xA+nI eq (− x) , n
n ∈ N0 .
(46)
Proof Using (3) yields
Dqk eq (−x ) = (−1)k k eq (− x) and
Dqn−k xA+nI = [A + nI]q [A + (n − 1)I]q . . . [A + (k + 1)I]q xA+kI A+(k+1)I q ; q n−k A+kI = x (1 − q)n−k which can be rewrite by using (31) as
Dqn−k xA+nI
=
A+I −1 A+I q ;q k q ;q n (1 − q)n−k
xA+kI .
From the Leibniz’s rule for the nth q-derivative of a product rule Koekoek and Swarttouw (1998)
Dqn {f (x)g(x)}
=
n n k=0
k
q
Dqn−k f
xq k
Dqk g (x),
n ∈ N0
and the properties of the matrix functional calculus, it follows that
Dqn
x
A+nI
A+I −1 A+I n ; q n kA+k 2 I k k q ;q k q A k n eq (− x) = x eq (− x) (−1) q x . n−k k q (1 − q) k=0
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 ∞ �f , g� = xA eq (− x)f (x)g(x)dq x (47) 0
Let Pn (x) be a matrix polynomials for n ≥ 0. We say that the sequence {Pn (x)}n≥0 is an orthogonal matrix polynomials sequence with respect to the inner product , provided for all nonnegative integers n and m
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1. Pn (x) is a matrix polynomial of degree n with non-singular leader coefficient. 2. �Pn (x), Pm (x)� = 0 for all n � = m. 3. Pn (x), Pn (x) is invertible for n ≥ 0. Let us assume that n ∈ N0, be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all 0 ≤ k ≤ n, then the first and second conditions for the orthogonality of the q-Laguerre matrix polynomials come from Definition 4 and Theorem 3. The second condition, upon the inner product, states
L(A, ) (x; q), L(A, ) n m (x; q) = 0,
(48)
n �= m
Lemma 15 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all k ∈ N0 . Then, we get
n q 2 n L(A, ) (x; q), L(A, ) (x; q) = q nA I0 (A + nI) n n [n]q !
(49)
where
I0 (A + nI) =
∞
xA+nI eq (− x)dq x
(50)
0 (A, )
Proof Since the q-Laguerre matrix polynomials Ln (x; q) are polynomials of degree n with a non-singular leader coefficient, then there exist constants matrices Ck , k = 0, 1, 2, . . . , n such that n
(A, )
Ck Lk
k=0
(x; q) = xn I
(51)
where I is the identity matrix in Cr×r. In view of (48) and (51), we can deduce that
xk , L(A, ) (x; q) = 0, n
k = 0, 1, 2, . . . , n − 1.
Thus from (19), we obtain
(−1)n n q n(A+nI) xn , L(A, ) (x; q) L(A, ) (x; q), L(A, ) (x; q) = n n n [n]q !
(52)
(A, )
To evaluate the inner product �xn , Ln (x; q)�, let us suppose that ∞ In (A) = xA+nI eq (− x)L(A, ) (x; q)dq x, n ∈ N0 . n 0
Inserting the Rodrigues-type formula for the q-Laguerre matrix polynomials (46) yields ∞ [n]q !In (A) = xn Dqn xA+nI eq (− x) dq x. 0
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On q-integrating by parts, which states ∞ f (x)(Dq g)(x)dq x = {f (x)g(x)}∞ − 0 0
∞
g(qx)Dq f (x)dq x,
0
we find
∞ [n]q !In (A) = xn Dqn−1 xA+nI eq (− x) 0 ∞ n−1 n−1 A+nI − [n]q x Dq x eq (− x) (qx)dq x 0
which can be rewritten by Rodrigues-type formula (46) as
∞ (A+I, ) [n]q !In (A) = [n − 1]q ! xA+(n+1)I eq (− x)Ln−1 (x; q) 0 ∞ (A+I, ) A+I A+nI − [n]q !q x eq (− qx)Ln−1 (qx; q)dq x. 0
By using Lemma 2, we obtain ∞ (A+I, ) A+I In (A) = −q xA+nI eq (− qx)Ln−1 (qx; q)dq x. 0
Using the q-analogue of the integration theorem by change of variable from qx to x yields ∞ (A+I, ) xA+nI eq (− x)Ln−1 (x; q)dq x = −q −n In−1 (A + I) In (A) = −q −n 0
which leads to
In (A) = (−1)n q −
n(n+1) 2
I0 (A + nI).
Hence, from (52), we have the desired results.
˜ > 0, then we Theorem 16 Let us assume that A ∈ Cr×r satisfying the condition µ(A) have ∞ Ŵq (A) = Kq (A, ) xA−I eq (− x)dq x (53) 0
where Ŵq (A) is the q-gamma matrix function defined by Salem (2012) as
Ŵq (A) =
1 1−q
xA−I Eq (−xq)dq x
(54)
0
and I−A
Kq (A, ) = (1 − q)
(− (1 − q); q)∞
−q ;q (1 − q)
−q I−A eq (− q ) ;q (1 − q) ∞ ∞
A
(55)
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Proof Let the function ∞ Ŵ˜ q (A) = xA−I eq (− x)dq x 0
and let
B˜ q (A, n) =
∞·(1−q)
0
− xq A+nI ; q (− x; q)∞
∞ A−I
x
dq x,
n∈N
It easy to show that
lim B˜ q (A, n) = (1 − q)A Ŵ˜ q (A)
(56)
n→∞
From the definition of q-derivative (3), we get
Dq
− xq A+nI ; q (− x; q)∞
∞ A+nI
x
= [A + nI]q
− xq A+(n+1)I ; q (− x; q)∞
∞ A+(n−1)I
x
On q-integrating by parts with using the results obtained in Lemma 2 and the above result, we obtain recursive formula
− xq A+nI ; q ∞ A+nI (qx) Dq B˜ q (A, n + 1) = x dq x (− x; q)∞ 0 ∞·(1−q) − xq A+nI ; q ∞ A−I x dq x = −q n [A + nI]−1 [−n] q q (− x; q)∞ 0 ˜ = [n]q [A + nI]−1 n∈N q Bq (A, n), n [A + nI]−1 q q
∞·(1−q)
−n
which can be read as
−1 B˜ q (A, n) = (q; q)n−1 q A+I ; q B˜ q (A, 1), n−1
n∈N
Also the function B˜ q (A, 1) can be computed as
B˜ q (A, 1) = [A]−1 q = Replacing
[A]−1 q
∞·(1−q)
Dq
0
− xq A ; q ∞ A x dq x (− x; q)∞ ∞·(1−q)
− xq A ; q ∞ A x (− x; q)∞
0
x by (1 − q)q −n yields
B˜ q (A, 1) =
A [A]−1 lim q (1 − q) n→∞
Let the function (1 − q)A−I Kq (A, ) = lim
n→∞
q
q −nA
−nA
− (1 − q)q A−nI ; q ∞ − (1 − q)q −n ; q ∞
−1 − (1 − q)q A−nI ; q ∞ − (1 − q)q −n ; q ∞
(57)
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Using (29) followed by (23) when n = k , we can deduce that I−A
Kq (A, ) = (1 − q)
(− (1 − q); q)∞
−q ;q (1 − q)
∞
eq
−q I−A ;q − q (1 − q) ∞ A
which concludes that
Kq (A, )B˜ q (A, 1) = [A]−1 q (1 − q)
(58)
In view of (56)–(58), we obtain
−1 Kq (A)Ŵ˜ q (A) = (q; q)∞ q A ; q (1 − q)I−A ∞
An important relation for the q-gamma matrix function was obtained by Salem (2012) as
−1 Ŵq (A) = (q; q)∞ q A ; q (1 − q)I−A , ∞
q −k �∈ σ q A , k ∈ N0
which reveals that
Kq (A)Ŵ˜ q (A) = Ŵq (A) This completes the proof. The results proved in this section can be summarized in the following theorem: Theorem 17 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all k ∈ N0 , then the q-Laguerre matrix pol(A, ) ynomials sequence {Ln (x; q)}n≥0 is an orthogonal matrix polynomials sequence with respect to the inner product
q n2 n (A, ) (A, ) q nA Kq−1 (A + (n + 1)I, )Ŵq (A + (n + 1)I)δmn . Ln (x; q), Lm (x; q) = [n]q !
(59)
Conclusion ˜ > −1 In our work, we introduce the q-Laguerre matrix polynomials (19) hold for µ(A) 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. Acknowledgements The author is looking forward to taking the opportunity to express his sincere gratitude to the anonymous referees for their valuable comments and suggestions that will be suggested. Competing interests The author declare that he has no competing interests. Received: 4 September 2015 Accepted: 18 April 2016
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References Constantine AG, Muirhead RJ (1972) Partial differential equations for hypergeometric functions of two argument matrix. J Multivar Anal 3:332–338 Defez E, Jódar L (1998) Some applications of the Hermite matrix polynomials series expansions. J Comput Appl Math 99:105–117 Dunford N, Schwartz J (1956) Linear operators, part I. Interscience, New York Duran AJ, Grunbaum FA (2005) A survey on orthogonal matrix polynomials satisfying second order differential equations. J Comput Appl Math 178:169–190 Exton H (1977) On a basic analogue of the generalised Laguerre equation. Funkc Ekvacioj 20:1–8 Gantmacher FR (1998) The theory of matrices, vol 1. American Mathematical Society, Providence Gasper G, Rahman M (2004) Basic hypergeometric series. Cambridge University Press, Cambridge Higham NJ (2008) Function of matrices. Theory and computation. SIAM, Philadelphia Higueras I, Garcia-Celaeta B (1999) Logarithmic norms for matrix pencil. SIAM J Matrix Anal 20:646–666 Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge Jódar L, Company R, Navarro E (1994) Laguerre matrix polynomials and system of second-order differential equations. Appl Numer Math 15:53–63 Jódar L, Sastre J (2001) The growth of Laguerre matrix polynomials on bounded intervals. Appl Math Lett 14:21–26 Jódar L, Sastre J (2004) Asymptotic expressions of normalized Laguerre matrix polynomials on bounded intervals. Util Math 65:3–31 Koekoek R (1992) Generalizations of a q-analogue of Laguerre polynomials. J Approx Theory 69(1):55–83 Koekoek R, Swarttouw RF (1998) The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report No. 98-17, Delft University of Technology, Delft Moak DS (1981) The q-analogue of the Laguerre polynomials. J Math Anal Appl 81:20–47 Salem A (2012) On a q-gamma and a q-beta matrix functions. Linear Multilinear Algebra 60(6):683–696 Salem A (2014) The basic Gauss hypergeometric matrix function and its matrix q-difference equation. Linear Multilinear Algebra 62(3):347–361 Sastre J, Defez E, Jódar L (2006) Laguerre matrix polynomial series expansion: theory and computer applications. Math Comput Model 44:1025–1043 Sastre J, Defez E (2006a) On the asymptotics of Laguerre matrix polynomials for large x and n. Appl Math Lett 19(8):721–727 Sastre J, Jódar L (2006b) On the asymptotics of Laguerre matrix polynomials. Util Math 70:71–98 Sastre J, Jódar L (2006) On Laguerre matrix polynomials series. Util Math 71:109–130
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Open Access
RESEARCH
The q‑Laguerre matrix polynomials Ahmed Salem* *Correspondence: [email protected] Department of Basic Science, Faculty of Information Systems and Computer Science, October 6 University, Sixth of October City, Egypt
Abstract 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. Keywords: q-Laguerre matrix polynomials, q-Gamma matrix function, Matrix functional calculus, Three terms recurrence relation, Rodrigues-type formula Mathematics Subject Classification: 33D05, 33D45, 15A16
Background The study of functions of matrices is a very popular topic in the Matrix Analysis literature. Some basic references are Gantmacher (1998), Higham (2008) and Horn and Johnson (1991). The subject of the orthogonal polynomials cuts across a large piece of mathematics and its applications. Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949. In the last two decades this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of n (Duran and Grunbaum 2005). The second-order matrix differential equations of the form xY ′′ (x) + (A + I − xI)Y ′ (x) + nY (x) = 0,
n ∈ N0 , A, Y (x) ∈ Cr×r
(1)
have been introduced and investigated in Jódar et al. (1994), where Cr×r denotes the vector space containing all square matrices with r rows and r columns with entries in the complex number C, ∈ C and x is a real number. The explicit expression for the nth (A, ) Laguerre matrix polynomial Ln (x) which is a solution of the second order matrix differential equation (1), has the form
L(A, ) (x) = n
n (−1)k k k (A + I)n (A + I)−1 k x , k!(n − k)! k=0
R( ) > 0
(2)
© 2016 Salem. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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where (A + I)n = (A + I)(A + 2I) . . . (A + (n − 1)I), n ∈ N and (A + I)0 = 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 Jódar and Sastre (2001, 2004), Sastre and Jódar (2006a, b), Sastre and Defez (2006), Sastre et al. (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
(Dq f )(x) =
f (x) − f (qx) , (1 − q)x
(3)
q �= 1, x �= 0
which becomes the same as ordinary differentiation in the limit as q → 1. We shall use the q-analogue of the product rule
(4)
(Dq fg)(x) = f (x)(Dq g)(x) + (Dq f )(x)g(qx).
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
L(α) n (x; q)
=
q α+1 ; q
n
(q; q)n
k n −n k(k−1) q ; q k q 2 q n+α+1 x , [k]q ! q α+1 ; q k k=0
α > −1, n ∈ N0
(5)
for 0 < q < 1, where [a]q = (1 − q a )/(1 − q), (a; q)n is the q-shifted factorial defined as
(a; q)n =
n−1 k=0
(1 − aq k ),
n∈N
with (a; q)0 = 1
and [n]q ! is the q-factorial function defined as
[n]q ! = [n]q [n − 1]q . . . [1]q ,
n∈N
with [0]q ! = 1.
The q-Laguerre polynomials (5) appeared as a solution of the second order q-difference equation
xDq2 y(x) + [α + 1]q − q α+2 x (Dq y)(qx) + [n]q q α+1 y(qx) = 0,
α > −1.
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).
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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
x(Dq2 Y )(x) + [A + I]q − x q A+2I (Dq Y )(qx) + [α]q CY (qx) = 0
(6)
where , α ∈ C and A, C and Y(x) are square matrices in Cr×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
||A|| = max x� =0
√ ||Ax||2 : ∈ σ (A∗ A) = max ||x||2
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 CN ×N such that σ (A) ⊂ �, then from the properties of the matrix functional calculus (Dunford and Schwartz 1956), it follows that f (A)g(A) = g(A)f (A). The logarithmic norm of a matrix A is defined as (Sastre and Defez 2006)
�I + hA� − 1 = max z : z ∈ σ A + A∗ /2 . h h→0
µ(A) = lim
˜ such that Suppose the number µ(A) µ(A) ˜ = −µ(−A) = min z : z ∈ σ A + A∗ /2 .
By Higueras and Garcia-Celaeta (1999), it follows that �eAt � ≤ etµ(A) for t ≥ 0, we have µ(A) t , if t ≥ 1, A �t � ≤ (7) ˜ , if 0 ≤ t ≤ 1. t µ(A)
If A(k, n) and B(k, n) are matrices on CN ×N for n, k ∈ N0, it follows that (Defez and Jódar 1998) ∞ ∞
B(k, n) =
∞ n
B(k, n − k)
(8)
∞ n
B(k, n) =
∞ ∞
B(k, n + k).
(9)
n=0 k=0
n=0 k=0
and
n=0 k=0
n=0 k=0
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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
lim xα eq (−ax) = 0
(10)
x→∞
where eq (x) is the q-analogue of the exponential function defined as
eq (x) =
∞ xk 1 = , [k]q ! (x(1 − q); q)∞ k=0
|x| < (1 − q)−1 .
(11)
Proof Let f (x) = xα eq (−ax). Taking the limit of logarithm of the function f(x) as x → ∞ (11), gives
lim log f (x) = lim
x→∞
x→∞
α log x −
∞ k=0
log 1 + axq k
Taking n ∈ N0 such that n ≥ α yields
lim log f (x) = lim
x→∞
x→∞
log
xα n−1 k=0
1 + axq k
−
∞ k=n
log 1 + axq k = −∞
This means that limx→∞ f (x) = e−∞ = 0 which completes the proof. Lemma 2 Suppose that A ∈ Cr×r satisfying the condition
(12)
µ(A) ˜ > −1 and let be a complex number with R( ) > 0. Then it follows that
lim xA+I eq (−qx )Pn (x) = 0
(13)
lim xA+I eq (−qx )Pn (x) = 0
(14)
x→0
and x→∞
where Pn (x) is a matrix polynomials of degree n ∈ N0. Proof From (7), we get ˜ �xA+I � ≤ xµ(A)+1 ,
x→0
Since eq (−qx )Pn (x) is bounded as x → 0, it follows that (13) holds. From (7), we get
�xA+I � ≤ xµ(A)+1 ,
x→∞
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Let Pn (x) = a1 xn + a2 xn−1 + · · · + an and let 0 ≤ k ≤ n, then Lemma 1 gives
lim xµ(A)+k+1 eq (−qx ) = 0,
R( ) > 0
x→∞
it follows that
lim xA eq (−qx )Pn (x) = 0,
R( ) > 0
x→∞
which ends the proof. Theorem 3 Let m, n ∈ N0, A ∈ Cr×r satisfying the condition (12) and C ∈ Cr×r is invertible and depends on A. Let Ym and Yn are solutions of matrix q-difference equation (6) corresponding to αm and αn respectively, then we get ∞ xA eq (−qx )Ym (qx)Yn (qx)dq x = 0, m �= n (15) 0
where the q-integral is the inverse of q-derivative (3) defined as
0
∞
f (x)dq x = (1 − q)
∞
k=−∞
qk f qk
Proof By virtue of (4), the matrix q-difference equation (6) can be read as
Dq xA+I eq (−qx )(Dq Y )(x) (x) + [α]q xA Ceq (−qx )Y (qx) = 0
(16)
Since Ym and Yn are solutions of (16) corresponding to αm and αn respectively, then we can easily obtain
[αm ]q − [αn ]q xA Ceq (−qx )Ym (qx)Yn (qx) = Ym (qx) Dq xA+I eq (−qx )(Dq Yn )(x) (x) − Dq xA+I eq (−qx )(Dq Ym )(x) (x)Yn (qx) = Dq xA+I eq (−qx )(Dq Yn )(x)Ym (x) − xA+I eq (−qx )Yn (x)(Dq Ym )(x) (x)
On q-integrating both sides from 0 to ∞ and by Lemma 2 and hypothesis αm � = αn yields ∞ ([am ]q − [an ]q )C xA eq (−qx )Ym (qx)Yn (qx)dq x = 0, m �= n 0
which ends the proof. Now, let us suppose that the solution of (6) has the form
Y (x) =
∞ k=0
ak xk ,
ak ∈ Cr×r ,
where 0r×r is the null matrix in Cr×r.
a0 �= 0r×r
(17)
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To determine the matrices ak . Taking formal q-derivatives (3) of Y (x), it follows that
(Dq Y )(x) =
∞ k=0
[k + 1]q ak+1 xk ,
and
∞ Dq2 Y (x) = [k + 1]q [k + 2]q ak+2 xk k=0
Substituting into (6) would yield ∞ k=0
∞ [k + 1]q [k + 2]q ak+2 xk+1 + [A + I]q − x q A+2I [k + 1]q ak q k xk k=0
+ [α]q C
∞ k=0
ak q k xk = 0
Equating the coefficients of xk , k ∈ N0 would yield
[A + I]q a1 + [α]q Ca0 = 0 and
[k]q + [A + I]q q k [k + 1]q ak+1 − q k [k]q q A+I − [α]q C ak = 0,
k∈N
which can be read as
[A + kI]q [k]q ak − q k−1 [k − 1]q q A+I − [α]q C ak−1 = 0,
(18)
k∈N
For existence of the second order q-difference equation, we seek the sufficient condition
C = q A+I ,
and α = n
for some non-negative integer n
We have to suppose that q −k � ∈ σ (q A ), k ∈ N0 to ensure that the relevant (I − q A+kI ) exists. Therefore, (18) gives
ak =
[k − n − 1]q A+(n+1)I [A + kI]−1 ak−1 , q q [k]q
k∈N
which leads to
ak =
q −n ; q
k
k q 2 k
[k]q !
q A+I ; q
−1 k
q (A+(n+1)I)k a0 A+I
A+I
n n which reveals that a0 = (q(q;q);q) and Letting the boundary condition Y (0) = (q(q;q);q) n n so we can seek the following definition for the q-Laguerre matrix function which verified the Eq. (6)
Definition 4 Let n ∈ N0, be a complex number with R( ) > 0 and A ∈ Cr×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
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L(A, ) (x; q) = n
n −n q ; q kq
k 2 k x k
[k]q !(q; q)n
k=0
q A+I ; q
−1 k
q A+I ; q
n
q (A+(n+1)I)k .
(19)
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 ˜ > −1 but Jódar et al. (1994) proved that the Laguerre matrix poly(19) hold for µ(A) nomials (2) hold for R(z) > −1 for all z ∈ σ (A) which equivalently β(A) > −1 where β = min{R(z) : z ∈ σ (A)}. It is worth noting that the important relation between β(A) ˜ ˜ , which states β(A) > µ(A) (Salem 2012). Therefore, the definition of the and µ(A) ˜ > −1. Laguerre matrix polynomials (2) can be extended for µ(A)
Generating functions The basic hypergeometric series is defined as Gasper and Rahman (2004)
a1 ,a2 ,...,ar r φs b1 ,b2 ,...,bs ; q, z
= r φs (a1 , a2 , . . . , ar ; b1 , b2 , . . . , bs ; q, z) =
∞ n s−r+1 (a1 , a2 , . . . , ar ; q)n zn, (−1)n q 2 (q, b1 , b2 , . . . , bs ; q)n
(20)
n=0
for all complex variable z if r ≤ s, 0 < |q| < 1 and for |z| < 1 if r = s + 1, where
(a1 , a2 , . . . , ar ; q)n = (a1 ; q)n (a2 ; q)n . . . (ar ; q)n Notice that the q-shifted function (a; q)n has the summation Koekoek and Swarttouw (1998)
(a; q)n = where
n
k q
n n k=0
k
q
k q 2 (−a)k
(21)
is the q-binomial coefficients defined as
−n n q ;q k n (q; q)n = (−1)k q kn− 2 . = (q; q)k (q; q)n−k (q; q)k k q
(22)
Also it has the well-known identities
(a; q)n−k
q k (a; q)n = 1−n −1 q − (q a ; q)k a
(a; q)n =
(a; q)∞ , (aq n ; q)∞
k −nk 2 ,
a �= 0,
k = 0, 1, 2, . . . n
(23)
and
n ∈ N0 .
(24)
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The q-shifted factorial matrix function was defined in Salem (2012) as
(A; q)0 = I,
(A; q)n =
n−1 k=0
I − Aq k ,
(25)
n∈N
and satisfies
(A; q)∞ = lim
n→∞
n−1 k=0
I − Aq k =
∞
I − Aq k =
k=0
∞ (−1)k q k=0
k 2
(q; q)k
Ak .
(26)
Furthermore, if A < 1 and q −k � ∈ σ (A), k ∈ N0, the infinite product (26) converges invertibly and
(A; q)−1 ∞ =
∞ k=0
Ak . (q; q)k
(27)
In Salem (2012) a proof of the matrix q-binomial theorem can be found ∞ (B; q)k k=0
(q; q)k
Ak = (A; q)−1 ∞ (AB; q)∞ ,
(28)
for all commutative matrices A, B ∈ Cr×r and q −k � ∈ σ (A), k ∈ N0. For complete this section, we need the following: Lemma 6 Let A be a square matrix inCr×r and n ∈ N0. Then, we have the following three identities ••
••
n
(A; q)n = (A; q)∞ Aq ; q
−1 ∞
=
n n k=0
k
q
k q 2 (−A)k ,
�A� < 1
(29)
where q −m
(A; q)−1 n
�∈ σ (A), m = n, n + 1, . . . , and its reciprocal ∞ n (q n ; q)k k = = (A; q)−1 Aq ; q A , �A� < 1 ∞ ∞ (q; q)k k=0
where q −k
•• Aq k ; q
(30)
�∈ σ (A), k ∈ N0. Also the identity
n−k
= (A; q)−1 k (A; q)n ,
k = 0, 1, . . . , n
holds for q −r � ∈ σ (A), r = 0, 1, . . . , k.
(31)
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Proof Let A < 1 and q −m � ∈ σ (A), m = n, n + 1, . . ., then we have
(A; q)n = =
n−1 k=0 ∞ k=0
1 − Aq k
1 − Aq k
= (A; q)∞
∞
k=0
∞
1 − Aq k
−1
k=n
1 − Aq k
∞ k=n
1 − Aq k+n
−1
1 − Aq k
−1
−1 = (A; q)∞ Aq n ; q ∞
Using (28) and (23) yields (29). The relation (30) comes immediately from (29) and (28). Also (31) can be easily obtained. Lemma 7 Let A and C are matrices inCr×r such that q −n � ∈ σ (C) for all n ∈ N0 and a ∈ C, the matrix functions 0 φ1 (−; C; q, A)
=
∞ q n(n−1) n=0
(q; q)n
n (C; q)−1 n A
(32)
and 0 φ2 (−; C, a; q, A)
=
3n(n−1) ∞ (−1)n q 2 n (C; q)−1 n A (q; q)n (a; q)n
(33)
n=0
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)
�A−1 − B−1 � ≤
�B−1 �2 �A − B� , 1 − �B−1 ��A − B�
(34)
one gets
−1 −1 � I − qn C � = � I − qn C − I + I� −1 − I� + 1 ≤ � I − qn C q n �C� +1 ≤ 1 − q n �C� ln �C� 1 . , n>− = 1 − q n �C� ln |q|
If we take
F (n) =
1, −1 n−1 k �, k=0 � I − q C
n = 0, n ≥ 1,
(35)
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and by the relation (35), we get
∞ n(n−1) ∞ (C; q)−1 q n(n−1) F (n) q n �0 φ1 (−; C; q, A)� ≤ An ≤ �A�n . (q; q)n (q; q)n n=0
n=0
Using the ratio test and the perturbation lemma (34), one finds
q n(n+1) F (n + 1)(q; q) �A�n+1 n lim n→∞ q n(n−1) F (n)(q; q)n+1 �A�n �(I − q n C)−1 � 2n q �A� n→∞ 1 − q n+1
= lim
≤ lim n→∞
q 2n �A� = 0. 1 − q n+1 (1 − q n �C�)
Thus, the matrix power series (32) is absolutely convergent. Similarly, (33) can be proved. This ends the proof. Since the function 1 φ1 (a; 0; q, z) is analytic for all complex numbers a and z, the matrix functional calculus tells that the matrix function 1 φ1 (a; 0; q, A) is also convergent for all complex number a and for all matrices A ∈ Cr×r. Lemma 8 Let A be matrix inCr×r such that q −n � ∈ σ (C) for all n ∈ N0 and a be a complex number. We have the transformation 1 φ1 (a; 0; q, A)
(36)
= (A; q)∞0 φ1 (−; A; q, aA)
Proof The relation (21) can be exploited to prove the transformation 1 φ1 (a; 0; q, A)
=
∞ (−1)n q
n 2 (a; q)n
(q; q)n
n=0
n
A
=
∞ (−1)n q n=0
n 2
(q; q)n
n
A
n n k=0
k
q
k q 2 (−a)k
Using the relations (9) and (26) lead to n+k k
+ ∞ ∞ 2 n + k (−1)n+k q 2 1 φ1 (a; 0; q, A) = (q; q)n+k k q
(−a)k
An+k
n=0 k=0
= =
∞ q k(k−1) ak k=0 ∞ k=0
(q; q)k
q k(k−1) ak (q; q)k
k
A
∞ (−1)n q n=0
n 2 +nk
(q; q)n
An
Ak (Aq k ; q)∞
Inserting the relation (30) into the above relation gives 1 φ1 (a; 0; q, A)
= (A; q)∞
This ends the proof.
∞ q k(k−1) ak k=0
(q; q)k
Ak (A; q)−1 k = (A; q)∞0 φ1 (−; A; q, aA)
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Theorem 9 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all k ∈ N0. The q-Laguerre matrix polynomials have the generating functions ∞
1 A+I t = L(A, ) (x; q)t n , 1 φ1 − x(1 − q); 0; q, q n (t; q)∞
(37)
n=0
∞ −1 (t; q)∞ tq −A−I ; q q −n(A+I) L(A, ) (x)t n 0 φ1 (−; t; q, − tx(1 − q)) = n ∞
(38)
n=0
∞
−1 1 A+I ; q, −q A+I (1 − q) xt = q A+I ; q L(A, ) (x; q)t n 0 φ1 −; q n n (t; q)∞
(39)
n=0
for all |t| < 1 and
(t; q)∞0 φ2 −; q A+I , t; q, − tx(1 − q)q A+I
=
∞ n=0
n −1 n q A+I ; q L(A, ) (x)t n (40) (−1) q 2 n n
for all t ∈ C. Proof The left hand side of (37) can be rewritten by means of using the transformation (36) as follows
1 A+I t 1 φ1 − x(1 − q); 0; q, q (t; q)∞ 1 A+I = ; q 0 φ1 −; tq A+I ; q, − xt(1 − q)q A+I tq ∞ (t; q)∞ ∞ (−1)n q n(n−) n xn t n A+I −1 n(A+I) 1 A+I tq ;q tq ;q q , = n ∞ (t; q)∞ [n]q !
LHS =
n=0
|t| < 1
Using (30) followed by (28) give
∞ (−1)n q n(n−1) n xn t n tq A+(n+1)I ; q ∞ n(A+I) LHS = q [n]q ! (t; q)∞ n=0 ∞ ∞ (−1)n q n(n−1) n xn t n q A+(n+1)I ; q k k n(A+I) t q = [n]q ! (q; q)k n=0
k=0
In view of (8), we get
LHS =
∞ n=0
t
n
n (−1)k q k(k−1) k xk k=0
[k]q !(q; q)n−k
q A+(k+1)I ; q
n−k
q k(A+I)
Inserting (23) and (31) with taking into account the definition of q-Laguerre matrix polynomials (19) to obtain the right hand side of (37). Similarly, we can find (38) with
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−A−I � < 1 which is equivalent to noting that the convergence of (tq −A−I ; q)−1 ∞ needs �tq A+I µ(A)+1 ˜ |t| < �q �≤q < 1. In order to prove (39), we have
1 A+I ; q, −q A+I (1 − q) xt 0 φ1 −; q (t; q)∞ ∞ ∞ t n (−1)n q n(n−1) n xn t n A+I −1 (A+I)n q ;q q = n (q; q)n [n]q ! =
n=0 ∞
n=0
tn
n=0
n k=0
(−1)k q k(k−1) k xk A+I −1 (A+I)k q q ;q k [k]q !(q; q)n−k
Using the well-known identity (23) to obtain (39). (40) is similar to (37) and (38). Corollary 10 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×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
L(A, ) (x; q) = n
n (q −n ; q)k (− x(1 − q); q)k
(q; q)k
k=0
q (A+(n+1)I)k .
(41)
Proof The generating function (37) can be expanded as
1 A+I t 1 φ1 − x(1 − q); 0; q, q (t; q)∞ n ∞ ∞ n n t (−1) q 2 (− x(1 − q); q)n (A+I)n n = q t (q; q)n (q; q)n n=0
n=0
k n ∞ k (−1) q 2 (− x(1 − q); q)k (A+I)k = q tn (q; q)k (q; q)n−k
= =
n=0
n=0
∞
n (q −n ; q)k (− x(1 − q); q)k
n=0
∞
tn
n=0
(q; q)k (q; q)n
q (A+(n+1)I)k
L(A, ) (x; q)t n . n
n=0
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 −n n = (A; q)n1 φ1 q −n ; A; q, xAq n (42) 2 φ1 q , x; 0; q, Aq which tends to the transformation (36) as n → ∞.
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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 ∈ Cr×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 (A, ) [n + 1]q Ln+1 (x; q) + xq A+(2n+1)I − q[n]q − [A + (n + 1)I]q L(A, ) (x; q) n (43) (A, ) + [A + nI]q Ln−1 (x; q) = 0, n ∈ N. Proof Let the matrix-valued function
F (x, t, A) =
1 A+I t 1 φ1 − x(1 − q); 0; q, q (t; q)∞
(44)
Using Jackson q-derivatives operator (3) and the q-analogue of the product rule (4) give
(Dq F )(t) =
1 A+I t 1 φ1 − x(1 − q); 0; q, q (1 − q)(t; q)∞ q A+I A+I − qt 1 φ1 − x(1 − q); 0; q, q (1 − q)(qt; q)∞ + x(1 − q)1 φ1 − x(1 − q); 0; q, q A+I q 2 t
Inserting the above relation into the generating function (37) with taking the q-derivative of the right hand side yields ∞ n=0
∞ n A+(n+1)I (x; q)t − q (x; q)t n+1 1 − q A+(n+1)I L(A, ) 1 − q L(A, ) n n n=0
− x(1 − q)q A+I =
∞ n=0
∞
L(A, ) (x; q)q 2n t n n
n=0
(A, )
(1 − q n+1 )Ln+1 (x; q)t n − q
∞ n=0
(A, )
(1 − q n+1 )Ln+1 (x; q)t n+1
Equating to the zero matrix the coefficient of each power t n it follows that (A, )
L1
(A, ) (x; q) = [A + I]q − xq A+I L0 (x; q)
and
(A, ) 1 − q A+(n+1)I L(A, ) (x; q) − 1 − q A+nI qLn−1 (x; q) n
− x(1 − q)q A+(2n+1)I L(A, ) (x; q) n (A, ) = 1 − q n+1 Ln+1 (x; q) − 1 − q n qL(A, ) (x; q). n
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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 ∈ Cr×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 n
(A, )
q i Li
i=0
(x; q) = Ln(A+I, ) (x; q),
n ∈ N0 .
(45)
Proof It is not difficult, by using (19) to see that the q-Laguerre matrix polynomials satisfy the forward shift operator (A+I, )
L(A, ) (x; q) − L(A, ) (qx; q) = − x(1 − q)q A+I Ln−1 n n
(qx; q)
which is equivalent to (A+I, )
Dq L(A, ) (x; q) = − q A+I Ln−1 n
(qx; q).
By iteration this process k-times, we can get the relation
(xq k ; q), k = 0, 1, . . . , n; Dqk L(A, ) (x; q) = (−1)k k q k(A+kI) L(A+kI, ) n n−k
n ∈ N0 .
When k = n, we obtain
Dqn L(A, ) (x; q) = (−1)n n q n(A+nI) , n
n ∈ N0
which can be also obtained from nth term of (19) with fact that Dqn xn = [n]q !. It is easy to show that
(1 − t)F (x, t, A + I) = F (x, qt, A) From (37), we can deduce that ∞ n=0
Ln(A+I, ) (x; q)t n −
∞ n=0
Ln(A+I, ) (x; q)t n+1 =
∞
q n L(A, ) (x; q)t n n
n=0
which gives (A+I, )
q n L(A, ) (x; q) = L(A+I, ) (x; q) − Ln−1 n n
(x; q).
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.
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Theorem 14 (Rodrigues-type formula) Let be a complex number with R( ) > 0 and A ∈ Cr×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
[n]q !xA eq (− x)L(A, ) (x; q) = Dqn xA+nI eq (− x) , n
n ∈ N0 .
(46)
Proof Using (3) yields
Dqk eq (−x ) = (−1)k k eq (− x) and
Dqn−k xA+nI = [A + nI]q [A + (n − 1)I]q . . . [A + (k + 1)I]q xA+kI A+(k+1)I q ; q n−k A+kI = x (1 − q)n−k which can be rewrite by using (31) as
Dqn−k xA+nI
=
A+I −1 A+I q ;q k q ;q n (1 − q)n−k
xA+kI .
From the Leibniz’s rule for the nth q-derivative of a product rule Koekoek and Swarttouw (1998)
Dqn {f (x)g(x)}
=
n n k=0
k
q
Dqn−k f
xq k
Dqk g (x),
n ∈ N0
and the properties of the matrix functional calculus, it follows that
Dqn
x
A+nI
A+I −1 A+I n ; q n kA+k 2 I k k q ;q k q A k n eq (− x) = x eq (− x) (−1) q x . n−k k q (1 − q) k=0
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 ∞ �f , g� = xA eq (− x)f (x)g(x)dq x (47) 0
Let Pn (x) be a matrix polynomials for n ≥ 0. We say that the sequence {Pn (x)}n≥0 is an orthogonal matrix polynomials sequence with respect to the inner product , provided for all nonnegative integers n and m
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1. Pn (x) is a matrix polynomial of degree n with non-singular leader coefficient. 2. �Pn (x), Pm (x)� = 0 for all n � = m. 3. Pn (x), Pn (x) is invertible for n ≥ 0. Let us assume that n ∈ N0, be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all 0 ≤ k ≤ n, then the first and second conditions for the orthogonality of the q-Laguerre matrix polynomials come from Definition 4 and Theorem 3. The second condition, upon the inner product, states
L(A, ) (x; q), L(A, ) n m (x; q) = 0,
(48)
n �= m
Lemma 15 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all k ∈ N0 . Then, we get
n q 2 n L(A, ) (x; q), L(A, ) (x; q) = q nA I0 (A + nI) n n [n]q !
(49)
where
I0 (A + nI) =
∞
xA+nI eq (− x)dq x
(50)
0 (A, )
Proof Since the q-Laguerre matrix polynomials Ln (x; q) are polynomials of degree n with a non-singular leader coefficient, then there exist constants matrices Ck , k = 0, 1, 2, . . . , n such that n
(A, )
Ck Lk
k=0
(x; q) = xn I
(51)
where I is the identity matrix in Cr×r. In view of (48) and (51), we can deduce that
xk , L(A, ) (x; q) = 0, n
k = 0, 1, 2, . . . , n − 1.
Thus from (19), we obtain
(−1)n n q n(A+nI) xn , L(A, ) (x; q) L(A, ) (x; q), L(A, ) (x; q) = n n n [n]q !
(52)
(A, )
To evaluate the inner product �xn , Ln (x; q)�, let us suppose that ∞ In (A) = xA+nI eq (− x)L(A, ) (x; q)dq x, n ∈ N0 . n 0
Inserting the Rodrigues-type formula for the q-Laguerre matrix polynomials (46) yields ∞ [n]q !In (A) = xn Dqn xA+nI eq (− x) dq x. 0
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On q-integrating by parts, which states ∞ f (x)(Dq g)(x)dq x = {f (x)g(x)}∞ − 0 0
∞
g(qx)Dq f (x)dq x,
0
we find
∞ [n]q !In (A) = xn Dqn−1 xA+nI eq (− x) 0 ∞ n−1 n−1 A+nI − [n]q x Dq x eq (− x) (qx)dq x 0
which can be rewritten by Rodrigues-type formula (46) as
∞ (A+I, ) [n]q !In (A) = [n − 1]q ! xA+(n+1)I eq (− x)Ln−1 (x; q) 0 ∞ (A+I, ) A+I A+nI − [n]q !q x eq (− qx)Ln−1 (qx; q)dq x. 0
By using Lemma 2, we obtain ∞ (A+I, ) A+I In (A) = −q xA+nI eq (− qx)Ln−1 (qx; q)dq x. 0
Using the q-analogue of the integration theorem by change of variable from qx to x yields ∞ (A+I, ) xA+nI eq (− x)Ln−1 (x; q)dq x = −q −n In−1 (A + I) In (A) = −q −n 0
which leads to
In (A) = (−1)n q −
n(n+1) 2
I0 (A + nI).
Hence, from (52), we have the desired results.
˜ > 0, then we Theorem 16 Let us assume that A ∈ Cr×r satisfying the condition µ(A) have ∞ Ŵq (A) = Kq (A, ) xA−I eq (− x)dq x (53) 0
where Ŵq (A) is the q-gamma matrix function defined by Salem (2012) as
Ŵq (A) =
1 1−q
xA−I Eq (−xq)dq x
(54)
0
and I−A
Kq (A, ) = (1 − q)
(− (1 − q); q)∞
−q ;q (1 − q)
−q I−A eq (− q ) ;q (1 − q) ∞ ∞
A
(55)
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Proof Let the function ∞ Ŵ˜ q (A) = xA−I eq (− x)dq x 0
and let
B˜ q (A, n) =
∞·(1−q)
0
− xq A+nI ; q (− x; q)∞
∞ A−I
x
dq x,
n∈N
It easy to show that
lim B˜ q (A, n) = (1 − q)A Ŵ˜ q (A)
(56)
n→∞
From the definition of q-derivative (3), we get
Dq
− xq A+nI ; q (− x; q)∞
∞ A+nI
x
= [A + nI]q
− xq A+(n+1)I ; q (− x; q)∞
∞ A+(n−1)I
x
On q-integrating by parts with using the results obtained in Lemma 2 and the above result, we obtain recursive formula
− xq A+nI ; q ∞ A+nI (qx) Dq B˜ q (A, n + 1) = x dq x (− x; q)∞ 0 ∞·(1−q) − xq A+nI ; q ∞ A−I x dq x = −q n [A + nI]−1 [−n] q q (− x; q)∞ 0 ˜ = [n]q [A + nI]−1 n∈N q Bq (A, n), n [A + nI]−1 q q
∞·(1−q)
−n
which can be read as
−1 B˜ q (A, n) = (q; q)n−1 q A+I ; q B˜ q (A, 1), n−1
n∈N
Also the function B˜ q (A, 1) can be computed as
B˜ q (A, 1) = [A]−1 q = Replacing
[A]−1 q
∞·(1−q)
Dq
0
− xq A ; q ∞ A x dq x (− x; q)∞ ∞·(1−q)
− xq A ; q ∞ A x (− x; q)∞
0
x by (1 − q)q −n yields
B˜ q (A, 1) =
A [A]−1 lim q (1 − q) n→∞
Let the function (1 − q)A−I Kq (A, ) = lim
n→∞
q
q −nA
−nA
− (1 − q)q A−nI ; q ∞ − (1 − q)q −n ; q ∞
−1 − (1 − q)q A−nI ; q ∞ − (1 − q)q −n ; q ∞
(57)
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Using (29) followed by (23) when n = k , we can deduce that I−A
Kq (A, ) = (1 − q)
(− (1 − q); q)∞
−q ;q (1 − q)
∞
eq
−q I−A ;q − q (1 − q) ∞ A
which concludes that
Kq (A, )B˜ q (A, 1) = [A]−1 q (1 − q)
(58)
In view of (56)–(58), we obtain
−1 Kq (A)Ŵ˜ q (A) = (q; q)∞ q A ; q (1 − q)I−A ∞
An important relation for the q-gamma matrix function was obtained by Salem (2012) as
−1 Ŵq (A) = (q; q)∞ q A ; q (1 − q)I−A , ∞
q −k �∈ σ q A , k ∈ N0
which reveals that
Kq (A)Ŵ˜ q (A) = Ŵq (A) This completes the proof. The results proved in this section can be summarized in the following theorem: Theorem 17 Let n ∈ N0 , be a complex number with R( ) > 0 and A ∈ Cr×r satisfying the conditions (12) and q −k � ∈ σ (q A ) for all k ∈ N0 , then the q-Laguerre matrix pol(A, ) ynomials sequence {Ln (x; q)}n≥0 is an orthogonal matrix polynomials sequence with respect to the inner product
q n2 n (A, ) (A, ) q nA Kq−1 (A + (n + 1)I, )Ŵq (A + (n + 1)I)δmn . Ln (x; q), Lm (x; q) = [n]q !
(59)
Conclusion ˜ > −1 In our work, we introduce the q-Laguerre matrix polynomials (19) hold for µ(A) 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. Acknowledgements The author is looking forward to taking the opportunity to express his sincere gratitude to the anonymous referees for their valuable comments and suggestions that will be suggested. Competing interests The author declare that he has no competing interests. Received: 4 September 2015 Accepted: 18 April 2016
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