 Advances in Pure Mathematics, 2011, 1, 315-321 doi:10.4236/apm.2011.16057 Published Online November 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM A New Extension of Humbert Matrix Function and Their Properties Ayman Shehata, Mohamed Abul-Dahab* 1Department of Mat hem at i cs, Faculty of Science, Assiut University, Assiut, Egypt 2Department of Mat hem at i cs, Faculty of Science, South Valley University, Qena, Egypt E-mail: *mamabuldahab@yahoo.com Received August 15, 2011; revised October 22, 2011; accepted October 30, 2011 Abstract This paper deals with the study of the composite Humbert matrix function with matrix arguments ,ABJz. The convergence and integral form this function is established. An operational relation between a Humbert matrix function and Kummer matrix function is studied. Also, integral expressions of this relation are de-duced. Finally, we define and study of the composite Humbert Kummer matrix functions. Keywords: Hypergeometric Matrix function, Humbert Matrix Function, Kummer Matrix Function, Integral Representations 1. Introduction Special matrix functions appear in the literature related to statistics [1-4] and more recently in connection with matrix analogues of Laguerre, Hermite and Legendre differential equations and the corresponding poly- nomial families [5-7]. The connection between the Humbert matrix function and modified Bessel matrix function has been established in [8,9]. In recent papers [10,11], we defined and studied the Humbert matrix functions. The Kummer’s confluent hypergeometric function belongs to an important class of special func- tions of the mathematical physics with a large number of applications in different branches of the quantum mecha- nics atomic physics, quantum theory, nuclear physics, quantum electronics, elasticity theory, acoustics, theory of oscillating strings, hydrodynamics, random walk theory, optics, wave theory, fiber optics, electromagnetic field theory, plasma physics, the theory of probability and the mathematical statistics, the pure and applied mathematics in [3,4,12-14]. Recently, an extension to the Kummer matrix function of complex variable is appeared in . The first author has earlier studied the certain Kummer matrix function of two complex variables under certain differential and integral operators . The primary goal of this paper is to consider a new system of matrix functions, namely the composite Humbert matrix func- tion, Humbert Kummer matrix function and composite Humbert Kummer matrix function. The paper is organized as follows: Section 2 is define and study of the composite Humbert matrix function. The convergence and integral form is established. In Section 3 an operational relation between a Humbert matrix function and Kummer matrix function is given. Integral expressions of Humbert Kummer matrix func- tions are deduced. In Section 4 we defined and studied of the composite Humbert Kummer matrix functions. Throughout this paper 0 will denote the complex plane. A matrix is a positive stable matrix in DPNNC if >0Re for all where PP is the set of all eigenvalues of and its two-norm denoted by P202=,supxPxPx where for a vector in yNC, 122=Tyyy is the Euclidean norm of . yLet P and P be the real numbers which were defined in  by  =max :,=min :.PRezzPPRezzP (1.1) If fz and gzz are holomorphic functions of the complex variable which are defined in an open set  of the complex plane and is a matrix in P A. SHEHATA ET AL. 316 NNC such that , then from the properties of the matrix functional calculus , it follows that P =.fPgPgPfP (1.2) Hence, if in QNNC=PQ QP is a matrix for which and if , then Q =.fPgQgQf P (1.3) The reciprocal Gamma function denoted by  1z 1=zz is an entire function of the complex va- riable . Then the image of acting on denoted by is a well-defined matrix. Further- more, if 1zPPinvertible fo  1PIPPnI P  =1lim1israll integer0.PnI n (1.4) The Pochhammer symbol or shifted factorial defined  by   0=1 ;1;=nPP nInP I (1.5) Jódar and Cortés have proved in  that  1! .PnPnPnQn (1.6) Let and be two positive stable matrices in PNNC. The gamma matrix function and the beta matrix function have been defined in  as follows P,BPQ0=e=exptPPttP10,=PIQ td;lnIPItIt. (1.7) and 1 dQIBPt t (1.8) The Schur decomposition of , was given by  in the form: P12; 0,!stPPrtts1()=0eert Ps and 121()=0ln;!srPPsPr nnn ns2. Composite Humbert M atrix Function Let us introduce the following notation (see ) 121212121212121212121212=,,,,=,=,=,,,,=,=,,,,=,=,=sskkkkssssssskkk ksskkk kskkk kkkkkzzz zAAA AAAA ABBB BBBB BAIA IAIAIBIBIBIBI   (2.1) and ,,, ,1122=,,,ABA BABABssJJJ J. Suppose that  11,302=3 ,;,;;=1,2,,27ABiiiAB iiiiiiiiizJzAI BIzFAIBI is  (2.2) are s Humbert matrix function with square complex matrices 12,,,sAAA and 12,,,sBB B of the same order N. Construct the function  11,30233=03 ,;,;27 ,ABABAB kIAB kIkzJzAIzFAIBIUz BI (2.3) where  113311=.3!kAB kIAB kI1AkI BkIUk   1. (1.9) This function, will be called the composite Humbert matrix function of several complex variables 12,,,szz z. Now we prove that the matrix power series (2.3) convergence for all 30z. Using the ratio test, we obtain  3( 1)113( 1)3( 1)33(1)() ()313131133()221=limsuplimsup 31!1( 1) .limsup 13!AB k IAB k IAB kIAB kIABkIkkAB kIAB kIAB kIkAk IBk IzUzRUz kAkI BkIzAkI BkIzzkk        Copyright © 2011 SciRes. APM A. SHEHATA ET AL.317 Note that if is large enough so that 1k 1>kA, then by perturma, , we can write bation lem1111=11AAk IIkk 111=11 1AIkkk A  (2.4) hence () 1111311=limsup 1.. 11..1 .. 11..|| ... 1kssssskkk AkAkBzkB   (2.5) For positive numbers 1Ri and positive integer , we can write k=,=1,2,2.iikk si (2.6) Substitute from (2.6) into (2.5) one gets 3( 1)3( 1)1=limsupAB kAB kIUz  3() 3IAB kIkAB kIRUz  Thus, the power series (2.3) is absolutely convergent for all =0.3kIBC,=0 !3nn11131333=0,,; =3;,,; ,27.nnn AnIAAkIAkIkBC zMABCzJ zzIBCAIIB ICzFIBCAIIBI CUz (3.4) Provided that 0.()>BC Where A, B and are matrices in C NNC such that (1)AkI, (1)kIC and (1)kIB This function, nction of are invertibwill be complex vale foriable r everythe Humsimplicity, we can write the Humber Kummer matrifunction in the form HKMF. We define the radius of regularity of the function ber For integer kKummer ma1. trix fucalled z.x (, , ;)MABCz in the form 133(() ()1=limsup limsuAkIAkIkIBB ICIkURk k This means MF is an entire function. Integral Expressions of Humbert Kummer Matrix Function In this section, we provide integral expressions of HKMF. Suppose that )1)) () (32133p1! ! = 0.limsu p!CAIIAkAAkIkkkkkkk  (3.5)  that the the HK(( )IBC and ()AI are matrices in NNC such that   =,,andare positive stable.IBCAI AIIBCIBC AIABC  (3.6) By (1.5) and (3.6) one gets . (3.7) By lemma 2 of  and (3.7), we see that   11111= 11BC kIBCA kAI ABCkIB CA B CAkI   1=kkIBC AII  11 AI IIBC 11()0=1 d.ABC IkBCCA kIttt1 (3.8) From relation (3.7) and (3.8), we get 1kIB CA B  1111()01d.kkABC IkBCIBC AIIBC AIABCttt   (3.9) Copyright © 2011 SciRes. APM A. SHEHATA ET AL. Copyright © 2011 SciRes. APM 319Hence, for 1,A ((=()(())))( )IBAIIBC C AI and )(( )IBC, ()AI, ()ABC are positive stable. Then for