A Certain Generalized Gamma Matrix Functions and Their Properties

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

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.

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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] Jódar and Cortés 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.

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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. B The factor exp( t ) in the integral (14) plays the role of a (iii)- This follows from (i) and (ii) when we replace A regularizer. For− this generalized we present difference by A and use the Eq.(21). formula, reflection formula, differential equation and −(iv)- Form Definition 3. 1. We have various particular cases. ∂ n Let us suppose that B is a matrix such that Γ (A,B) = ( 1)nΓ (A nI,B) ∂Bn { } − − (22) Re(z) > 0, for every eigenvalye z σ(B), µ(B) > 0, n = 0,1,2,... ∈ 1 Replacing A by A I in (i), we find 2 1 and let us denote √B = B = exp( 2 logB) the image of − 1 Γ Γ Γ 2 1 B (A 2I,B) (I A) (A I,B) (A,B)= 0, (23) the function z = exp( 2 logz) by the Riesz-Dunford − − − − − functional calculus, acting on the matrix B, where logz form (22) and (23), we obtain the required second-order denotes the principal branch of the complex logarithm differential equation. (see [8]). Then by the integral in (14) can be simplified in Definition 3.2. Let A and B be positive stable matrices in terms of the Macdonald matrix function to give N N C × such that for all integral n 0 satisfies the condition A ≥ Γ (A,B)= 2exp( lnB)K (2√B); µ(B) > 0, (15) 2 A A + nI and B + nI are invertible. (24) is also useful in the evaluation of certain Mellin and Then Γ (A,B) is invertible, its inverse coincides with 1 Laplace transforms. Γ − (A,B) and the generalized Psi (Digamma) matrix Theorem 3.1. Let A and B be positive stable matrices in function ψ(A,B) given in the formula N N C × , then each of the following properties holds true: ∂ ψ(A,B) := ln(Γ (A,B)) (i)Γ (A + I,B)= AΓ (A,B)+ BΓ (A I,B). ∂ A{ } (25) (ii)Γ A B exp AlnB Γ A B− ∂ ( , )= ( ) ( , ). 1 − − = Γ − (A,B) Γ (A,B) . (iii)Γ (I A,B)= exp( AlnB) Γ (A + I,B) AΓ(A,B) . ∂A{ } − − − ∂ 2U ∂U Γ It follows from the integral representation (14) of the (iv)B 2 + (I A) ∂ U = 0; U = (A,B). ∂ B − B − generalized Psi matrix function that Proof.(i)-Let M be Mellin transform operator as defined ∞ 1 A I It Bt 1 by ψ(A,B)= Γ − (A,B) t − (lnt)e− − − dt ∞ Z0 (26) A I µ µ M f (t);A := f (t)t − dt. (16) (B) 0, (A) 0. { } 0 ≥ ≥ Z Remark 3.1. (14) and (26) are matrix versions of Then Γ (A,B) is simply the Mellin transform of 1 generalized Gamma and Psi functions given in [6]. f (t) := exp[ It Bt− ] in A, thus − − Remark 3.2. (14) and (26) for B = 0 reduces Gamma and It Bt 1 Γ (A,B) := M e− − − ;A . Psi matrix functions in [16,18], respectively. { } Theorem 3.2. Let A and B be positive stable matrices in Exploiting the relationship CN N satisfying the condition (24) for all integral n 0. × ≥ M f ′(t);A = (A I)M( f (t)(A I)), (17) Then each of the following properties holds true: { } − − −

c 2015 NSP Natural Sciences Publishing Cor. 66 M. A. Abul-Dahab, A. k. Bakhet: A Certain Generalized Gamma Matrix Functions...

∞ (i)ψ(A,B)= Γ 1(A,B) The transformation s = et 1 in the second integral on the 0 − − s A ds right -hand side yields e− Γ (AR,B) (1 + s)− Γ (A,B(1 + s)) ; × − s µ µ (Bn) 0, (A) > 0. o ∞ AΓ Γ 1 ≥ ∞ tΓ AtΓ t (1 + s)− (A,B(1 + s)) − (A,B) ψ Γ 1 e− (A,B) e− (A,Be ) ds (ii) (A,B)= 0 − (A,B) t 1 e t dt; δ s − − − Z (34) µ(B) 0, Rµ(A) > 0. n o ∞ Γ (A,Bet )Γ 1(A,B)e tA ≥ ∞ 1 − − dt (iii)ψ(A,B)= lnA + Γ − (A,B) = t . 0 ln(1+δ ) 1 e Z − − Γ (A,B) Γ (A,BeR t ) At t 1 e t e− dt; × ( − − − ) From (33) and (34), we obtain the required relationship µ(B) 0,; µ(A) > 0. ∞ e t Γ A Bet Γ 1 A B e tA (iv)ψ( A≥,B)= lnB + ψ(A,B); µ(B) > 0. ψ − ( , ) − ( , ) − (A,B)= t dt. (35) ψ − Γ ψ Γ 0 t − 1 e (v) (A + I,B) (A + I,B) B (A I,B) (A + I,B) Z − Γ ψ Γ− −µ µ − = (A,B)+A (A,B) (A,B); (B) 0, (A) > 0. At ≥ e− (iii)- Adding and subtracting the factor t in the first term proof. (i)-Let us consider the double integral of the integrand in (ii), we find that ∞ ∞ t s t(1+s) A I Bt 1 e− − e− ψ Λ = t − e− − − dt ds. (27) (A,B) 0 0 s Z Z ∞ e t e At ∞ 1 If we integrate the double integral with respect to t, = − − − dt + t t (36) becomes Z0 Z0 Γ (A,Bet )Γ 1(A,B)e tAn − − e At dt. ∞ ∞ 1 e t − s A I (It+Bt 1) − − Λ = e− t − e− − dt − o Z0 Z0 The first on the right- hand side of (34) is the standard ∞ A I (t(1+s)I+Bt 1) ds integral representation of ln(A) (see [6]). This proves (iii). t − e− − dt . (28) − 0 s (iv)-Replacing A by A in (26), it follows that Z − From Definition 3.1. We obtain ∞ 1 (A+I) (tI+Bt 1) ψ( A,B)= Γ − ( A,B) t− lnte− − dt.(37) − − 0 ∞ Z Λ sΓ AΓ ds = e− (A,B) (1 + s)− (A,B(1 + s)) .(29) The substitutions t = Bτ ,dt = Bτ 2dτ in (37) yield 0 − s − − Z n o Integrating (27) with respect to s, we have ψ( A,B) − ∞ ∞ ∞ s ts 1 (38) 1 e e A 1 A I τI Bτ− Λ A I (It+Bt− ) − − =B− Γ − ( A,B) (lnB lnτ)τ − e− − dτ. = t − e− − ds dt. (30) − 0 − 0 0 s Z Z nZ o The inner integral in (30) is the integral representation of Using result (ii) in Theorem 3.1. We have lnt ([6], p. 24), we see that ∞ ψ( A,B) 1 ∂ Λ tA I lnte (It+Bt− )dt Γ A B (31) − ∞ = − − = ∂ ( , ). 1 A I τI Bτ 1 (39) 0 A =Γ − (A,B) (lnB lnτ)τ − e− − − dτ, Z 0 − Form (29) and (31), it follows that Z ∂ which is exactly the right-hand side of (iv)(v)- According (Γ (A,B)) ∂A to the relation (15) with straightforward computation ∞ shows that s A = e− Γ (A,B) (1 + s)− (32) ∞ 0 − A A I A 2I tI Bt 1 Z lnt t At − Bt − e− − − dt ds 0 − − Γ (A,B(1 + s)) . Z A/2 (40) s =2B KA(2√B), Γ 1 Multiplying both sides in (32) by − (A,B). Thus, the µ(B) > 0, proof of relation (i) is completed. (ii)-From result (i), we see that that is, ψ A B ( , ) Γ (A + I,B)ψ(A + I,B) AΓ(A,B)ψ(A,B) ∞ s − (41) e− Γ ψ Γ = lim ds B (A I,B). (A,B)= (A,B). δ 0 δ s (33) − − → Z ∞ h A 1 1 Γ 1 (1 + s) Γ (A,B(1 + s))Γ (A,B) Multiplying the above equation by A− − (A,B) and − − ds . − δ s rearranging the terms completes the proof of (v). Z i

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4 A generalized Pochhammer symbol 5 Concluding comments

The introduction of the generalized Gamma matrix The material developed in Sections 3 4 provides several Γ function (A,B) in Section 3 is useful in defining important properties of the generalized− Gamma matrix generalization of the Pochhammer symbol (A)n, n 1, Γ ≥ function (GGMF) (A,B), introduced in (14), under a (cf., e.g., [16,17,18]). The introduction of the concept and certain conditions on the matrices A and B. By using the notion of this symbol is destined to lead to the analytic GGMF, we have defined generalizations of Psi matrix study of a class of problems in engineering and other function and Pochhammer symbol. We have investigated sciences. As a matter of fact, several known properties of some properties of these generalized functions, most of the Pochhammer symbol are recovered from those of the which are analogous with the original matrix functions. generalized Pochhammer symbol introduced here. We Most of the special matrix functions of mathematical begin by introducing the generalized Pochhammer physics and engineering, such as hypergeometric, symbol under condition (24) can be written in the form Whittaker and modified Bessel matrix functions can be 1 (A,B)n = Γ (A + nI,B)Γ − (A); µ(B) > 0. (42) expressed in terms of generalized Pochhammer symbol. Which leads us easily to integral representation for the Therefore, the corresponding extensions of several other generalized Pochhammer symbol (A;B)n: familiar special matrix functions are expected to be useful ∞ and need to be investigated. 1 A+(n 1)I 1 (A;B)n =Γ − (A) t − exp (tI + Bt− )dt; 0 − Z (43) µ(B) > 0, µ(A + nI) > 0. 6 Acknowledgement Since the generalized Pochhammer symbol (A;B)n is related to the modified Bassel function of the third kind The authors are grateful to the anonymous referee for a from the relation (15), can be written as follows: careful checking of the details and for helpful comments A + nI (A;B) = 2Γ 1(A) exp( lnB) K (2√B) that improved this paper. n − 2 A+nI (44) , µ(B) > 0. Theorem 4.1. Let A and B be positive stable matrices in References CN N , such that satisfying the condition (24) and m,n × ∈ N0 := N 0. Then [1] G. Andrews, R. Askey and R. Roy, Special Functions, ∪ Cambridge University Press, New York, (1999). (A;B)n+m = (A)n(A + nI;B)m. (45) [2] E. T. Copson, An Introduction to the Theory of Functions of a proof. Form the relations (6) and (42), we find that Complex Variable, Oxford University Press, London, (1935). Γ 1 Γ [3] A. G. Constantine and R.J. Muirhead, Partial differential (A;B)n+m = − (A) (A + I(n + m),B) equations for hypergeometric functions of two argument =Γ (A + I(m + n),B)Γ (A + nI)Γ 1(A) matrix, J. Multivariate Anal., 3, (1972), 332-338. − (46) 1 [4] M. A. Chaudhry, N.M. Temme and E.J.M. Veling, Γ − (A + nI) × Asymptotic and closed form of a generalized incomplete =(A)n(A + nI;B)m. gamma function, J. Comput. Appl. Math., 67, (1996), 371- Thus, by applying the well-known properties of the 379. [5] M. A. Chaudhry and S.M. Zubair, Generalized incomplete Pochhammer symbol (A)n in (46) (see, for example, [16, 17,18,22]), it is fairly straightforward to derive the gamma functions with applications, J. Comput. Appl. Math., corresponding properties of the generalized Pochhammer 55, (1994), 99-124. [6] M. A. Chaudhry and S.M. Zubair, On a Class of symbol A;B as follows: ( )n Incomplete Gamma Functions with Applications, Chapman Corollary 4.1 Let k,l,m,n N0. Then each of the ∈ and Hall/CRC, Boca Raton, (2002). following identities holds true: [7] P. J. Davis, Leonhard Euler’s integral: a historical profile of (i)(A;B)n+m+l = (A)n(A + nI)m(A + (n + m)I;B)l, the gamma function, Amer. Math. Monthly., 66, (1959), 849- 869. (2m) A A+I [8] N. Dunford and J. Schwartz, Linear Operators., vol. I, (ii)(A;B)2m+l = 2 ( 2 )m( 2 )m(A + 2mI,B)l, Interscience, New York, (1957). [9] T.M. Flett, Diffzrential Analysis, Cambridge University (iii)(A + nI;B)n+l = (A + nI)n(A + 2nI;B)l 1 Press,(1980). = (A)2n[(A)n]− (A + 2nI,B)l, [10] I.I. Guseinov and B.A. Mamedov, Evaluation of incomplete (iv)(A + kmI;B) gamma functions using downward recursion and analytical kn+l relations, J. Math. Chem., 36, (2004), 341-346. = (A) [(A) ] 1(A + Ik(n + m);B) , k(m+n) km − l [11] I. I. Guseinova and B. A. Mamedov Unified treatment for the evaluation of generalized complete and incomplete (v)(A kmI)kn+l − km gamma functions, J. Comput .Appl. Math., 202, (2007), 435- = ( 1) (A)k(n m)(I A)km(A + k(n m);B)l. − − − − 439.

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[12] I. Higueras and B. Garcia-Celayeta, Logarithmic norms for M. A. Abul-Dahab matrix pencils, SIAM J. Matriz Anal., 20 (3), (1999), 646-666. M. A. Abul-Dahab earned [13] D.S. Jones, Diffraction by a waveguide of finite length, his M. Sc degree. In 2010 in Proc. Cambridge Philos. Soc., 48, (1952), 118-134. pure mathematics from South [14] A. T. James, Special Functions of Matrix and Single Valley University, faculty Argument in Statistics in Theory and Application of Special of science, Quna, Egypt. Functions, (Edited by R.A. Askey), pp. 497-520, Academic In 2013, he received his PhD Press, (1975). degree from South Valley [15] L. Jódar and J. Sastre, The growth of Laguerre matrix University, faculty of science, polynomials on bounded intervals, Appl. Math. Lett., 13, Quna, Egypt. His research (2000), 21-26. interests include Complex Analysis, Special Function and [16] L. Jódar and J.C. Cortés, Some properties of Gamma and Matrix Analysis. He has published research articles in Beta matrix functions, Appl. Math. Lett., 11, (1998), 89-93. [17] L. Jódar and J.C. Cortés, On the hypergeometric matrix reputed international journals of mathematical sciences. function, J. Comp. Appl. Math., 99, (1998), 205-217. He is referee and editor of mathematical journals. [18] Z. Kishka, A.Shehata and M. A. Abul-Dahab, On Humbert matrix functions and their properties, Afr. Mat., 24, (2013) 615-623. [19] K. Kobayashi, On generalized gamma functions occurring A. K. Bakhetis M. Sc. in diffraction theory, J. Phys. Soc. Jpn., 60, (1991), 1501-512. student in the Mathematics [20] K. Kobayashi, Plane wave diffraction by a strip: exact and Department, Faculty asymptotic solutions, J. Phys. Soc. Jpn., 60, (1991), 1891- of Science, Al-Azhar 1905. Assiut University, Assiut [21] H. M. Srivastava, R.K. Saxena and Chena Ram, A 71524, Egypt. unified presentation of the gamma-type functions occurring in diffraction theory and associated probability distributions, Appl. Math. Comput., 162, (2005), 931-947. [22] J. Sastre and L. Jódar, Asymptotics of the Modified Bessel and Incomplete Gamma Matrix Functions, Appl. Math. Lett., 16, (2003), 815-820.

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