On p and q-horn s Matrix Function of Two Complex Variables

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Alied Matheatics 437-44 doi:.436/a..3 Published Olie Deceber (htt://www.scirp.

1 Alied Matheatics doi:.436/a..3 Published Olie Deceber (htt:// O ad -or s Matrix Fuctio of Two Colex Variables Abstract Aya Shehata Deartet of Matheatics Faculty of Sciece Assiut Uiversity Assiut Egyt E-ail: drshehata6@yahoo.co Received Seteber 4 ; revised October 5 ; acceted Noveber 3 The ai ai of this aer is to defie ad study of a ew or s atrix fuctio say the ad -or s atrix fuctio of two colex variables. The radius of regularity o this fuctio is give whe the ositive itegers ad are greater tha oe a itegral reresetatio of ( A ABBCzw ; ; is obtaied recurrece relatios are established. Fially we obtai a higher order artial differetial euatio satisfied by the ad -or s atrix fuctio. Keywords: yergeoetric Matrix fuctios ad -or s Matrix Fuctio Cotiguous Relatios Matrix Fuctios Matrix Differetial Euatio Differetial Oerator. Itroductio May secial fuctios ecoutered i atheatical hysics theoretical hysics egieerig ad robability theory are secial cases of hyergeoetric fuctios []. yergeoetric series i oe ad ore variables occur aturally i a wide variety of robles i alied atheatics statistics [-4] ad oeratios research ad so o [5]. I [67] the hyergeoetric atrix fuctio has bee itroduced as a atrix ower series ad a itegral reresetatio. Moreover Jódar ad Cortés itroduced studied the hyergeoetric atrix fuctio F( ABCz ; ; the hyergeoetric atrix differetial euatio i [8] ad the exlicit closed for geeral solutio of it has bee give i [9]. Uadhyaya ad Dhai have earlier studied the geeralized or s fuctios of atrix arguets with real ositive defiite atrices as arguets [] ad this fuctio 7 also [] while the author has earlier studied the or s atrix fuctio of two colex variables uder differetial oerators [7]. I [ 3] extesio to the atrix fuctio fraework of the classical failies of -Kuer s atrix fuctios ad ad -Aell atrix fuctios have bee roosed. Our urose here is to itroduce ad study a extesio of the atrix fuctios of two variables. This aer is orgaized as follows: Sectio cotais the defiitio of the ad -or s atrix fuctio of two variables its radius of regularity ad itegral relatio of the ad -or s atrix fuctio is give. Soe atrix recu- rrece relatios are established i Sectio 3. Fially the effect of differetial oerator o this fuctio is ivestigated ad ad -or s atrix artial differetial euatio are obtaied i Sectio 4. Throughout this aer D will deote the colex lae cut alog the egative real axis. The sectru of a atrix A i C N N deoted by ( A is the set of its eigevalues of A. If A is a atrix i C its twoor deoted by A is defied by [4] Ax A su x x where for a vector y i N T C y y y is the Eu- clidea or of y. If f(z ad g(z are holoorhic fuctios of colex variables z defied i a oe set of the colex lae ad if A ad B are a atrix i C with ( A ad ( B also ad if AB BA the fro the roerties of the atrix fuctioal calculus [5] it follows that f( A g( B g( B f( A. (. The recirocal gaa fuctio deoted by ( z is a etire fuctio of the colex vari- ( z able z. The for ay atrix A i C NN the iage of ( z actig o A deoted by ( A is a welldefied Coyright SciRes.

2 438 atrix. Furtherore if A I is ivertible for every (. o egative iteger where I is the idetity atrix i C the ( A is ivertible its iverse coicides with ( A ad oe gets [8] ( A AA ( I ( A( I A I A A I. ( ( ; ;( Jódar ad Cortés have roved i [6] that A ( A li(![( A ]. (.3 (.4 Let P ad Q be two ositive stable atrices i C. The gaa atrix fuctio ( P ad the beta atrix fuctio BPQ ( have bee defied i [6] as follows ad t P I P I ( P Ilt ( P e t d t; t e (.5 PI QI BPQ ( t ( t d t. (.6 Let P ad Q be coutig atrices i C such that the atrices PI QI ad PQI are ivertible for every iteger. The accordig to [8] we have BPQ ( ( P ( Q PQ. (.7. Defiitio of ad -or s Matrix Fuctio Suose that ad are ositive itegers. The ad - or s atrix fuctio ( A A B B ; C ; z w of two colex variables is writte i the for where = ( AA BB ; Czw ; ( A - ( A' ( B ( B' [( C ] (!(! U ( V V ad ( A ( A ( B ( B [( C ]. (!(! (. For silicity we ca write the ( A ABBCzw ; ; i the for ; ; ( A+I A B B C i the for ( A ( ; ; A A B B C+I i the for ( C. We begi the study of this fuctio by calculatig its radius of regularity R of such fuctio for this urose we recall relatio (.3. of [78] ad keeig i id that. We defie the radius of re gularity of the fuctio ( A A B B ; C ; z w as where V li su R li su ( A ( A ( B ( B [( C ] (!(! A A B B ( ( A A ( A A ( B B ( B' lisu (!( (! (! (! (! (! (! (! C B [( C ] C (! (! (!(! ( C ( A ( A' ( B ( B (!(!(! li su (!(! A AC BB ( ;. Usig Stirlig forula ad take is a ositive iteger the Coyright SciRes.

3 A AC BB ( (!(!(! lisu [ ( ] ( R (!(! 439 ( π( π (!(!(! ( ( e li su li su e (!(! π π e e li su. ( ( Suarizig the followig result has bee established. As a coclusio we get the followig result. Theore.. Let A A B B ad C be atrices i C such that C I are ivertible for all iteger. The the ad -or s atrix fuctio is a etire fuctio i the case that at least oe of the itegers ad are greater tha oe. If the the fuctio is covergece i z r w s ad ( r s i [59]. Itegral for of the ad -or Matrix Fuctio Suose that A ad C are atrices i the sace C of the suare colex atrices such that AC CA A C ad C A are ositive stable atrices. By (.3 (.4 ad (.7 oe gets C A I ( C ( A C I A( - I ( A ( C-A-I ( A CA ( C t t d t. Substitutig fro (. ad (. we see that = ( AA BB ; Czw ; ( A ( B ( B' (!(! A( I CAI ( A ( C A ( C t ( t dt ( A ( CA ( C A-I CAI t t 3F A B B zt w ( ;-; d t. (. Therefore the followig result has bee established. Theore.. Let A A B B ad C be atrices i C. The the ad -or s atrix fuctio of two colex variables satisfies the followig itegral for where ( AA BB ; Czw ; = ( A' ( C A' ( C A-I CAI t t ( ;-; d 3F A B B zt w t (.3 ( A ( B ( B. 3F ( A BB ;-; ztw = ( zt w (!(! 3. Matrix Recurrece Relatios Soe recurrece relatio are carried out o the ad -or s atrix fuctio. I this coectio the followig cotiguous fuctios relatios follow directly by icreasig or decreasig oe i origial relatio ad (A+ ( A I ( A' ( B ( B' [( C ] (!(! A A I ( ( ( A I ( A ( B ( B [( C ] (!(! A A I U ( ( ( ( A I ( A ( B ( B [( C ] (3. (A (!(! Siilarly ( A I[( A ( I] U ( z w. (3. Coyright SciRes.

4 44 ( A+ A AIU ( z w ( A AI A I U ( z w ( B+ B BIU ( z w ( B+ B BIU ( z w ( B BI B I U ( z w ( B (C+ C CIU ( z w B I B I U ( z w (C CI C I U ( z w. ( The ad -or s Matrix Fuctio uder the Differetial Oerator Cosider the differetial oerator D o the ad -or s atrix fuctio of two colex variables defied i [7 7] as d d D otherwise where d z ad d w. This oerator has the z w roerty Dz w ( z w. For the ad -or s atrix fuctio the followig relatios hold ad ( DI A A I A ( A d ( A ( A ( B ( B [( C ] (!(! (4. A di - A I (!(! A ( A. By the sae way we have ( A ( A ( B ( B [( C ] di B B ( B di B B ( B ( dici C I ( C. Fro (4. (4. ad (4.3 we get A ( A A A B d A B A ( ' ( B A ( A A A B d A ( ' ( B. A B Fro (4. (4.3 ad (4.4 we have ( A A ABC d C ( A A C (4. (4.3 (4.4 ( C I ( B ( B AB C d C I ( B ( B. Also fro (4. (4.3 ad (4.4 we see that A C A C I C ( A C I ( C B B B B A ( ( B B B ( B B ( B ACBB A ( ( '. (4.5 (4.6 Now we aed this sectio by itroducig the dif- feretial oerator d z ad d w to the e- z w tire fuctios i successive aer as follows; Coyright SciRes.

5 44 d d d d d d d d ( A ( A ( B ( B [( C ]... (!(! ( A ( A ( B ( B [( C ]... (!(! ( A ( A ( B ( B [( C ]... (!(! ( A. ( A ( B ( B [( C ].. (!(! ( A ( A ( B ( B [( C ] ( A ( A ( B ( B [( C ] (!(! (!(! ( A ( A ( B ( B [( C ] ( A ( A ( B ( B [( C ] (!(! (!(! z ( A ( A ( B ( B [( C ] ( A( I( A I C I (!(! w AI ( B I( B I ( A ( A ( B ( B ( C (!(! z w AA( ( A A B B; C ; z w A I ( A A B B ; C; C BB i.e. d d d d d d d d z w AA( ( A A B B; C ; z w A I ( A A B B ; C; C BB We ca writte the ( A A B B ; C ; z w the d d d d d I C I d d d d d I I C I ( A ( A ( B ( B' [( C ] ( ( A ( A' ( B ( (!(! z w DI A di A d di A DI A d I d I A di B' [( C ] (!(!. Therefore the followig result has bee established. Theore 4.. Let A A B B ad C be atrices i C. The the ( A A B B ; C ; z w is a solutio for the followig differetial euatio Coyright SciRes.

6 44 dd d... d di CI dd d... d di I. z w ( DI A( di A d( di A' ( DI A di ( di A d I ( Ackowledgeets The Author exresses his sicere areciatio to Dr. M. S. Metwally (Deartet of Matheatics Faculty of Sciece (Suez Suez Caal Uiversity Egyt for his kid iterest ecourageets hel suggestios coets ad the ivestigatios for this series of aers. 6. Refereces []. M. Srivastava ad P. W. Karlsso Multile Gaussia yergeoetric Series Ellis-orwood Chichester 985. [] A. G. Costatie ad R. J. Mairhead Partial Differetial Euatios for yergeoetric Fuctios of Two Arguet Matrices Joural of Multivariate Aalysis Vol. No doi:.6/47-59x(79-6 [3] A. M. Mathai A adbook of Geeralized Scial Fuctios for Statistical ad Physical Scieces Oxford Uiversity Press Oxford 993. [4] A. M. Mathai Jacobias of Matrix Trasforatios ad Fuctios of Matrix Arguet World Scietific Publishig New York 997. [5]. M. Srivastava ad. L. Maocha A Treatise o Geeratig Fuctios Ellis orwood New York 984. [6] M. T. Mohaed ad A. Shehata A Study of Aell s Matrix Fuctios of Two Colex Variables ad Soe Proerties Joural Advaces ad Alicatios i Matheatical Scieces Vol. 9 No [7] A. Shehata A Study of Soe Secial Fuctios ad Polyoials of Colex Variables Ph.D. Thesis Assiut Uiversity Assiut 9. [8] L. Jódar ad J. C. Cortés O the yergeoetric Matrix Fuctio Joural of Coutatioal ad Alied Matheatics Vol. 99 No doi:.6/s377-47( [9] L. Jódar ad J. C. Cortés Closed for Geeral Solutio of the yergeoetric Matrix Differetial Euatio Matheatical ad Couter Modellig Vol. 3 No doi:.6/s (87-4 [] L. M. Uadhyaya ad. S. Dhai Geeralized or s Fuctios of Matrix Arguets Bulleti Pure ad Alied Scieces: Sectio E. Matheatics ad Statistics Vol. 9E No [] L. M. Uadhyaya A Suatio Forula for a or s Double yergeoetric Fuctio-II Bulleti Pure ad Alied Scieces: Sectio E. Matheatics ad Statistics Vol. E No [] Z. M. G. Kishka M. A. Salee S. Z. Radi ad M. Abul- Dahab O the ad -Aell Matrix Fuctio Southeast Asia Bulleti of Matheatics Vol [3] M. S. Metwally O -Kuers Matrix Fuctio of Colex Variable uder Differetial Oerators ad Their Proerties Southeast Asia Bulleti of Matheatics Vol [4] G. Golub ad C. F. Va Loa Matrix Coutatios The Johs okis Uiversity Press Baltiore 989. [5] N. Duford ad J. Schwartz Liear Oerators Part I Itersciece New York 955. [6] L. Jódar ad J. C. Cortés Soe Proerties of Gaa ad Beta Matrix Fuctios Alied Matheatics Letters Vol. No doi:.6/s (9739- [7] K. A. M. Sayyed Basic Sets of Polyoials of Two Colex Variables ad Covergece Proerties Ph.D. Thesis Assiut Uiversity Assiut 975. [8] K. A. M. Sayyed M. S. Metwally ad M. T. Mohaed Certai yergeoetric Matrix Fuctio Scietiae Matheaticae Jaoicae Vol. 69 No htt:// [9] A. Erdélyi W. Magus F. Oberhettiger ad G. Tricoi igher Tascedetal Fuctios McGraw-ill Book Co. New York Vol Coyright SciRes.

GENERALIZED KERNEL AND MIXED INTEGRAL EQUATION OF FREDHOLM - VOLTERRA TYPE R. T. Matoog

GENERALIZED KERNEL AND MIXED INTEGRAL EQUATION OF FREDHOLM - VOLTERRA TYPE R. T. Matoog Assistat Professor. Deartet of Matheatics, Faculty of Alied Scieces,U Al-Qura Uiversity, Makkah, Saudi Arabia Abstract: