mathematics Article Extended k-Gamma and k-Beta Functions of Matrix Arguments Ghazi S. Khammash 1, Praveen Agarwal 2,3,4 and Junesang Choi 5,* 1 Department of Mathematics, Al-Aqsa University, Gaza Strip 79779, Palestine; [email protected] 2 Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India; [email protected] 3 Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan 4 Harish-Chandra Research Institute Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211019, India 5 Department of Mathematics, Dongguk University, Gyeongju 38066, Korea * Correspondence: [email protected] Received: 17 August 2020; Accepted: 30 September 2020; Published: 6 October 2020 Abstract: Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered. Keywords: k-gamma function; k-beta function; gamma function of a matrix argument; beta function of matrix arguments; extended gamma function of a matrix argument; extended beta function of matrix arguments; k-gamma function of a matrix argument; k-beta function of matrix arguments; extended k-gamma function of a matrix argument; extended k-beta function of matrix arguments 1. Introduction and Preliminaries Special functions of matrix arguments have appeared in statistics, theoretical physics, theory of group representations and number theory (see, e.g., [1–3]). Special functions of a matrix argument were investigated in the study of spherical functions on certain symmetric spaces and multivariate analyses in statistics (see [4]). Certain properties of the gamma function of a matrix argument and beta function of matrix arguments and hypergeometric functions of matrix arguments have been studied (see, e.g., [5–9]). Various k-special functions such as the k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated (see, e.g., [10–15]). Some k-special functions of matrix arguments such as k-gamma function of a matrix argument, k-beta function of matrix arguments and k-hypergeometric functions of matrix arguments have been introduced and investigated (see [16,17]). In this paper, we extend the k-gamma and k-beta functions of matrix arguments and investigate some properties of the extended functions. + In what follows, we shall denote by Z, N, R, R , and C the classes of integers, positive integers, real numbers, positive real numbers, and complex numbers, respectively. Also put N0 := N [ f0g, − + + r×r Z0 := Z n N, and R0 := R [ f0g. For r 2 N, let C denote the set of all r by r matrices of × × which the entries are in C. Let $(A) be the set of all eigenvalues of A 2 Cr r. For A 2 Cr r, × let a(A) := max f<(z) j z 2 $(A)g and b(A) := min f<(z) j z 2 $(A)g. Let A be a matrix in Cr r such that <(z) > 0 (z 2 $(A)). (1) Mathematics 2020, 8, 1715; doi:10.3390/math8101715 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1715 2 of 13 × For A 2 Cr r, its 2-norm is denoted by kAxk kAk = sup 2 , x6=0 kxk2 r H 1/2 H where for any vector y 2 C , kyk2 = y y is the Euclidean norm of y. Here y denotes the Hermitian matrix of y. × If R is a matrix in Cs s which satisfies <(w) > 0 for all w 2 $(R), then G(R) is well-defined as follows: Z ¥ G(R) = e−u uR−I du, uR−I := exp (R − I) ln u, (2) 0 where I is the identity matrix of order s (see, e.g., [6–8,18]). Also, throughout this paper, let I denote the identity matrix corresponding to a square matrix of any order. If a(w) and b(w) are analytic functions of the complex variable w, which are defined in an open × set Q of the complex plane, and R is a matrix in Cs s such that $(R) ⊂ Q, one finds from the properties × of the matrix functional calculus that a(R) b(R) = b(R) a(R) (see [19] (p. 550)). Hence, if S in Cs s is another matrix with $(S) ⊂ Q, such that RS = SR, then a(R)b(S) = b(S)a(R) (see [6,7]). Since the reciprocal Gamma function denoted by G−1(w) = 1/G(w) is an entire function of the complex variable × − w, for any R in Cs s, the Riesz–Dunford functional calculus reveals that the image of G 1(w) acting on − × R, denoted by G 1(R), a well-defined matrix (see [19] (Chapter 7)). Moreover, if T is a matrix in Cs s which gratifies T + nI is invertible for every integer n 2 N0, (3) then G(T) is invertible, its inverse coincides with G−1(T), and − − T(T + I) ··· (T + (n − 1)I)G 1(T + nI) = G 1(T)(n 2 N) (4) (see [20] (p. 253)). Under condition (3), from that, (4) can be written in the form − T(T + I) ··· (T + (n − 1)I) = G(T + nI) G 1(T)(n 2 N). (5) The Pochhammer symbol (or shifted factorial) (l)n is defined (for l, n 2 C) by ( G(l + n) 1 (n = 0; l 2 C n f0g) (l)n := = (6) G(l) l(l + 1) ··· (l + n − 1)(n = n 2 N; l 2 C), it being accepted conventionally that (0)0 = 1. Now, one applies the matrix functional calculus to this × function to find that, for any matrix R in Cs s, (R)n = R(R + I) ··· (R + (n − 1)I)(n 2 N), (R)0 = I. (7) Also, in view of (5), (7) can be expressed in terms of the Gamma function of the matrix argument: −1 (R)n = G(R + nI) G (R) (n 2 N0) . (8) Jódar and Cortés [6] (Theorem 1) proved the following limit expression of the Gamma function of matrix argument (cf. [21] (p. 2, Equation (6))): − G(P) = lim (n − 1)! (P) 1 nP (n 2 N), (9) n!¥ n × where P 2 Cr r satisfies (1). Mathematics 2020, 8, 1715 3 of 13 × If f (P) is well defined and S is an invertible matrix in Cr r, then [22] (p. 541) f SPS−1 = S f (P) S−1. (10) × Using the Schur decomposition of P 2 Cr r, it follows [22] (pp. 336, 556) that ( p ) r−1 kPk r tj tP t a(P) + e ≤ e ∑ t 2 R . (11) j=0 j! + Let x 2 C, k 2 R , and n 2 N. Then the k-Pochhammer symbol (x)n,k and the k-gamma function Gk are defined by (see [10,11]) (x)n,k = x(x + k)(x + 2k) + ··· (x + (n − 1)k), (x)0,k = 1 (12) and n x −1 n! k (nk) k − Gk(x) = lim x 2 C n kZ0 . (13) n!¥ (x)n,k The Eulerian integral representation of the k-gamma function Gk in (13) is given by (see [10,11]) Z ¥ k x−1 − t + Gk(x) = t e k dt k 2 R , <(x) > 0 . (14) 0 The k-gamma function Gk satisfies the following fundamental functional relation + − Gk(x + k) = x Gk(x) k 2 R , x 2 C n kZ0 . (15) The k-Pochhammer symbol (x)n,k in (12) can be expressed in terms of the k-gamma function Gk as follows: Gk(x + n k) + − (x)n,k = k 2 R , n 2 N0, x 2 C n kZ0 . (16) Gk(x) The beta function B(a, b) is defined by (see, e.g., [21] (p. 8, Equation (43))) 8 Z 1 > ta−1( − t)b−1 t ( f<( ) <( )g > ) <> 1 d min a , b 0 B(a, b) = 0 (17) G(a) G(b) − > a, b 2 C n Z . : G(a + b) 0 × Let P, Q be matrices in Cr r satisfying (1). Then the beta function B(P, Q) of matrix arguments is well defined as follows (see [6]): Z 1 B(P, Q) = tP−I (1 − t)Q−I dt. (18) 0 × Further, if P, Q are diagonalizable matrices in Cr r such that PQ = QP, then B(P, Q) = G(P) G(Q) G−1(P + Q). (19) × By application of the matrix functional calculus, the k-Pochhammar symbol of a matrix P in Cr r is defined as (see [16]) (P)n,k = P(P + kI)(P + 2kI) ··· (P + (n − 1)kI) (n 2 N) (20) Mathematics 2020, 8, 1715 4 of 13 + and (P)0,k = I (k 2 R ). The limit expression and integral form of the k-gamma function of a matrix × P in Cr r satisfying (1) are given as follows (see [16]) n P −1 −1 + Gk(P) = lim n! k (nk) k (P) k 2 R (21) n!¥ n,k and Z ¥ k P−I − t + Gk(P) = t e k dt k 2 R . (22) 0 r×r + If P is a matrix in C such that P + nkI is an invertible matrix for every n 2 N0 and k 2 R , −1 then Gk(P) is invertible, its inverse with Gk (P), one finds (see [16]) −1 + (P)n,k = Gk(P + nkI) Gk (P) k 2 R , n 2 N0 .