J. Ana. Num. Theor. 3, No. 1, 63-68 (2015) 63 Journal of Analysis & Number Theory An International Journal http://dx.doi.org/10.12785/jant/030110 A Certain Generalized Gamma Matrix Functions and Their Properties 1, 2 M. A. Abul-Dahab ∗ and A. K. Bakhet 1 Dept. of Math., Faculty of Science, South Valley University, Qena 83523, Egypt 2 Dept. of Math., Faculty of Science, Al-Azhar Assiut University, Assiut 71524, Egypt Received: 20 Sep. 2014, Revised: 13 Nov. 2014, Accepted: 10 Dec. 2014 Published online: 1 Jan. 2015 Abstract: The main aim of this paper is to present generalizations of Gamma and Psi matrix functions. Some different properties are established for these new generalizations. By means of the generalized Gamma matrix function, we introduced the generalized Pochhammer symbols and their properties. Keywords: Generalized Gamma matrix functions, Psi matrix functions, Generalized Pochhammer symbols. 1 Introduction Gamma functions [2,13,19,20,21]. Some fundamental properties of these functions were investigated in [19]. In The Gamma function Γ (z), defined for Re(z) > 0 by the [10,11] a new approach to the computation of the improper integral generalized complete and incomplete Gamma functions ∞ are proposed, which considerably improved its z 1 v Γ (z)= v − e− dv, capabilities during numerical evaluations in significant Z0 cases. Using binomial expansion theorem the generalized was introduced into analysis in the year 1729 by Leonard Gamma and incomplete Gamma functions are expressed Euler [7], while seeking a generalization of the factorial through the familiar Gamma and exponential integral n! for non-integral values of n, and subjected to intense functions. study by many eminent mathematicians of the nineteenth and early twentieth centuries and continues to interest the A generalization of a well-known special matrix present generation. The logarithmic derivative ψ of the functions, which extends the domain of that matrix Gamma function is known as the Psi or digamma functions, can be expected to be useful provided that the function, that is, it is given by important properties of the special matrix functions are carried over to the generalization in a natural and simple Γ ′ (x) manner. Of course, the original special matrix functions ψ(x)= , Γ (x) and its properties must be recoverable as a particular case of the generalization. Special functions of a matrix for positive real numbers x. The Gamma functions are one of the most important special functions and have many argument appear in the study of spherical functions on certain symmetric spaces and multivariate analysis in applications in many fields of science, for example, statistics, see [14]. Special functions of two diagonal analytic number theory, statistics and physics. The generalized Gamma functions are widely used in matrix argument have been used in [3]. Some properties the solution of many problems of wave scattering and of the Gamma and Beta matrix functions have recently been treated in [16,17]. An asymptotic expression of the diffraction theory [19,20]. For the properties of the incomplete Gamma matrix function and integral analogous extensions of the family of Gamma functions, we refer the reader to [4,5,6], and for a historical profile, expressions of Bessel matrix functions are given in [22]. we refer the reader to [7]. To date, a few various methods The primary goal of this paper is to consider a have been developed for the analysis of the generalized generalizations of Gamma and Psi matrix functions. ∗ Corresponding author e-mail: [email protected], [email protected] c 2015 NSP Natural Sciences Publishing Cor. 64 M. A. Abul-Dahab, A. k. Bakhet: A Certain Generalized Gamma Matrix Functions... Then, difference formula, reflection formula, recurrence 2 Preliminaries relations and differential equation are derived for generalizations of Gamma and Psi matrix functions. In this section, we shall adopt in this work a somewhat Finally, we define generalizations of Pochhammer different notation and facts from that used by previous symbols and some their properties. results as follows: 1 Throughout this paper, consider the complex space The reciprocal Gamma function denoted by Γ − (z)= CN N 1 × of complex matrices of common order N. A matrix Γ z is an entire function of the complex variable z. Then N N ( ) A is a positive stable matrix in C × if Re(λ) > 0 for all N N 1 for any matrix A in C × , the imageof Γ − (z) acting on A λ σ(A) where σ(A) is the set of all eigenvalues of A. Γ 1 ∈ denoted by − (A) is a well defined matrix. Furthermore, The two-norm of A, which will be denoted by A , is if defined by Ax A + nI is invertible for all integer n 0, (5) A = sup 2 , ≥ || || N N || || x=0 x 2 where I is the identity matrix in C × , then Γ (A) is || || 1 invertible, its inverse coincides with Γ − (A) and one gets N H 1 where for a vector y in C , y 2 = (y y) 2 is the Euclidean the formula [16] H || || norm of y and A denotes the Hermitian adjoint of A. We 1 (A)n = Γ (A + nI)Γ − (A); n 1;(A)0 = I. (6) denote by µ(A) the logarithmic norm of A, defined by [9, ≥ 12,22] J´odar and Cort´es have proved in [16] that H Γ 1 A A + A (A)= lim (n 1)![(A)n]− n . (7) µ (1) n ∞ (A)= max z : z eigenvalues of . → − { 2 } N N For any matrix A and in C × we use the following We denote by µ(A) the number relation due to [17], as follows ∞ A + AH A (A)n n µ (1 z)− = ∑ z ; z < 1. (A)= min ze: z eigenvalues of . (2) n! { 2 } − n=0 | | CN N Bye [12], it follows that Let A be two positive stable matrices in × . The Gamma matrix function Γ A and the Digamma matrix µ ( ) eAt et (A); t 0. function ψ(A) have been defined in [17,18] as follows ≤ ≥ ∞ Hence, t A I A I Γ (A)= e− t − dt, t − = exp((A I)lnt) (8) 0 − N 1 1 s Z µ − ( A N 2 t) etA et (A) ∑ ; t 0, and ≤ s! ≥ s=0 d 1 e ψ(A)= lnΓ (A)= Γ − (A)Γ ′ (A). (9) and dA Jodar and Sastre have proved asymptotic behavior of N 1 1 s µ − ( A N 2 lnn) Gamma matrix function in [15] by using nA n (A) ∑ ; n 1. ≤ s=0 s! ≥ γ ∞ A A 1 e AΓ (A)= e A Π (I + )e n − , (10) − n=1 n − If f (z) and g(z) are holomorphic functions of the γ h i complex variable z which are defined in an open set Ω of where is the Euler-Mascheroni constant. N N Some integral forms of the Bessel matrix functions and the complex plane and A is a matrix in C × such that σ(A) Ω, then from the properties of the matrix the modified Bessel matrix functions proved in [22] arethe functional⊂ calculus [8], it follows that following Γ 1 I 1 z A − (A + 2 ) 2 A I f (A)g(A)= g(A) f (A). (3) JA(z) = ( ) (1 t ) − 2 cos(zt)dt, (11) 2 √π 1 − Z N N − Hence, if B in C × is a matrix for which σ(B) Ω and ⊂ Γ 1 I 1 if AB = BA, then z A − (A + 2 ) 2 A I IA(z) = ( ) (1 t ) − 2 cosh(zt)dt (12) 2 √π 1 − f (A)g(B)= g(B) f (A). (4) Z− and N N Let A in C × , F(z) a matrix function, and let g(z) be a ∞ 2 1 z A z (A+I) positive scalar function. We say that F(z) behaves KA(z)= ( ) exp( t ) t− dt, (13) 2 2 0 − − 4t O(g(z),A) in a domain Ω, if F(z) M(A)g(z), for Z || || ≤ some positive constant M(A),z Ω, and F(z) commutes where KA(z), also known as modified Hankel matrix with A. ∈ function or Macdonald matrix function. c 2015 NSP Natural Sciences Publishing Cor. J. Ana. Num. Theor. 3, No. 1, 63-68 (2015) / www.naturalspublishing.com/Journals.asp 65 3 A generalized Gamma matrix functions between the Mellin transform of a function and derivative we see that It is possible to extend the classical Gamma function in 2 ( It Bt 1) (A I)Γ (A I,B)= M ( I + Bt− )e − − − ;A .(18) infinitely many ways. some of these extensions could be − − − { − } useful in certain types of problems. We define the Hence, generalized Gamma matrix function (GGMF) and the generalized Psi matrix function (GPMF) in Definition 3.1 (A I)Γ (A I,B)= Γ (A,B)+ BΓ (A 2I;B). (19) − − − − − and Definition 3.2. The generalized Gamma and Psi Replacing A by A = A + I in (19), we get the proof matrix function have several interesting properties. Some of(i). useful properties are listed below in Theorem 3.1 and Theorem 3.2 of this section. (ii)-Putting t = Bξ 1, in (14) yield Definition 3.1. Let A and B be positive stable matrices in − ∞ N N A I Bξ 1 ξ C × , then, the generalized Gamma matrix function Γ (A,B)= exp(AlnB) ξ ( + )e ( − + )dξ , (20) Γ − − (A,B) is defined by Z0 ∞ which is exactly A I (It+ B ) Γ (A,B) := t − e− t dt, Z0 (14) Γ ( A,B)= exp( AlnB)Γ (A,B), (21) A I − − t − = exp((A I)lnt). − as desired.