A Study of H -Function of Two Variables

Transcription

1 ISSN: (A ISO 97: 007 Certified Orgaizatio) Vol., Issue 9, Setember 01 A Study of -Fuctio of Two Variables Yashwat Sigh 1 armedra Kumar Madia Lecturer, Deartmet of Mathematics, Goermet College, Kaladera, Jaiur, Raastha, Idia 1 Lecturer, Deartmet of Mathematics, Seth Moti Lal (P.G.) College, Jhuhuu, Raastha, Idia Abstract: I the reset aer, the author has itroduced a ew secial fuctio amely -fuctio of two ariables i the literature of secial fuctios. The defiitio, coergece, asymtotic behaiors hae bee cosidered. Net, we obtai some roerties of -fuctio of two ariables. Later o, the author establish a series reresetatio for the -fuctio of two ariables. Some iterestig secial cases of -fuctio of two ariables by comarig the series reresetatio of -fuctio of two ariables ad -fuctio of oe ariable is also established. I the last, the author has gie two secial cases of -fuctio of two ariables which are ot the secial cases of -fuctio of two ariables. Key words: -fuctio, -fuctio of two ariables, -fuctio, Series reresetatio. (000 Mathematics Subect Classificatio: C70) I. INTRODUCTION The -fuctio occurrig i the aer will be defied ad rereseted by Buschma ad Sriastaa [1] as follows: where 1, 1, P M, N M, N ( a ; ; A ) N,( a ; ) N P P,, ( b, ) 1, M,( b, ; B ) M 1, z z ( ) M N 1 1 i 1 ( ) zd i (1.1) i ( b ) (1 a ) (1 b ) ( a ) M 1 N 1 B P Which cotais fractioal owers of the gamma fuctios. ere, ad throughout the aer a ( 1,..., ) ad b ( 1,..., ) are comle arameters, 0( 1,..., P), 0( 1,..., ) (ot all zero simultaeously) A (1.) ad eoets A ( 1,..., N) ad B ( N 1,..., ) ca take o o iteger alues. The followig sufficiet coditio for the absolute coergece of the defiig itegral for the -fuctio gie by equatio (1.1) hae bee gie by (Buschma ad Sriastaa). Coyright to IJIRSET

2 ISSN: (A ISO 97: 007 Certified Orgaizatio) Vol., Issue 9, Setember 01 M N P (1.) A B 0 ad 1 1 M 1 N 1 1 arg( z) (1.4) The behaior of the -fuctio for small alues of z follows easily from a result recetly gie by (Rathie [6],.06,eq.(6.9)). We hae MN, b P, z 0 z, mi Re, z 0 1N If we take A 1( 1,,..., N), B 1( M 1,..., ) i (1.1), the fuctio -fuctio []. Coyright to IJIRSET (1.5) MN, P, [.] reduces to the Fo s The followig series reresetatio for the -fuctio will be required i the sequel [see Rathie,.05-06,eq.(6.8)]: M, N P, M a, ; A, a, z b,, b, ; B M 1, N N 1, P 1, M M 1, N h, r h, r r b 1 a ( 1) z h1 r0 1 1 h hr, B P,, 1 b a r! h r h r h M 1 N 1 b r h. h A II. TE -FUNCTION OF TWO VARIABLES hr, (1.6) The -fuctio of two ariables will be defied ad rereseted i the followig maer:, 1:, :, a, ; A, c 1,, ; K, c 1 1,,, e 1,, E ; R, e 1,, E o m m 1,, y 1, q1:, q ;, q y y b, ; B, d,,, ;,,,, ; 1, q d 1 1, m L f m 1, q F f 1, m F S m1, q 1 1, ( ) ( ) y dd (.1) 4 L1 L

3 ISSN: (A ISO 97: 007 Certified Orgaizatio) Vol., Issue 9, Setember 01 1, 1 1 a A 1 1 q1 a A 1 b B K m 1 c d 1 1 q c 1 d 1 m1 m R 1 e E f F 1 1 q e E 1 f F 1 m1 L S ad y are ot equal to zero (real or comle), ad a emty roduct is iterreted as uity i, qi, i, m (.) (.) (.4) are o-egatie itegers such that 0, o m q ( i 1,,;,). All the i i a ( 1,,..., ), b ( 1,,..., q ), c ( 1,,..., ), d ( 1,,..., q ), 1 1 e ( 1,,..., ), f ( 1,,..., q ) are comle arameters. 0( 1,,..., ), 0( 1,,..., q ) (ot all zero simultaeously), similarly E 0( 1,,..., ), F 0( 1,,..., q ) (ot all zero simultaeously). The eoets K ( 1,,..., ), L ( m 1,..., q ), R ( 1,,..., ), S ( m 1,..., q ) ca take o o-egatie alues. The cotour L1 is i -lae ad rus from to i. The oles of K right ad the oles of 1 c ( 1,,..., ), 1 a A ( 1,,..., 1) i d ( 1,,..., m ) lie to the to the left of the cotour. For K ( 1,,..., ) ot a iteger, the oles of gamma fuctios of the umerator i (.) are coerted to the brach oits. Coyright to IJIRSET

4 ISSN: (A ISO 97: 007 Certified Orgaizatio) Vol., Issue 9, Setember 01 The cotour L is i -lae ad rus from to i. The oles of R right ad the oles of 1 e E ( 1,,..., ), 1 a A ( 1,,..., 1) i f F ( 1,,..., m ) lie to the to the left of the cotour. For R ( 1,,..., ) ot a iteger, the oles of gamma fuctios of the umerator i (.4) are coerted to the brach oits. The fuctios defied i (.1) is a aalytic fuctio of ad y, if 1 q1 q (.5) U q1 q (.6) V A E B F The itegral i (.1) coerges uder the followig set of coditios: 1 1 m q q1 (.7) L K m m q q1 (.8) A A F F S E R E B m arg, arg y (.9) The behaior of the -fuctio of two ariables for small alues of z follows as: [, y] 0( y ),ma, y 0 (.10) mi Re d 1 m For large alue of z, mi Re f 1m F ' ' (.11) [, y] 0, y,mi, y 0 (.1) c 1 e 1 ' ma Re K, ' ma Re 1 R 1 E (.1) Coyright to IJIRSET

5 ISSN: (A ISO 97: 007 Certified Orgaizatio) Vol., Issue 9, Setember 01 Proided that U 0 ad V 0. If we take K 1( 1,,..., ), L 1( m 1,..., q ), R 1( 1,,..., ), S 1( m 1,..., q ) i (.1), the -fuctio of two ariables reduces to -fuctio of two ariables due to [5]. III. SOME PROPERTIES OF -FUNCTION OF TWO VARIABLES 1. If we set 1 1 q1 0, the -fuctio of two ariables breaks u ito a roduct of two -fuctio of m, c, ; K, c 1,, m, e 1,, E ; R, e, 1, E 1,, q, q y d,, d, ;,,, ; 1, m L f m 1, q F f 1, m F S m 1, q. If 0, we the obtai. oe ariable amely a, ; A, c, ; K, c,, e, E ; R, e, E, ;,,,, ;,,,, ; 0, 1 : m, : m, 1, 1 1, 1, 1, 1, 1, q1:, q:, q y b B d d L f F f F S 1, q1 1, m m1, q 1, m m1, q a, ; A, c, ; K, c,, e, E ; R, e, E, ;,,,, ;,,,, ; 0, 1 : m, : m, 1, 1 1, 1, 1, 1, 1, q 1:, q :, q y b B d d L f F f F S 1, q1 1, m m 1, q 1, m m 1, q a, ; A, c, ; K, c,, e, E ; R, e, E b B d d L f F f F S 0, 1 : m, : m, 1 1, 1 1, 1, 1, 1, 1, q1:, q:, q 1, ;,,,, ;,,,, ; y 1, q1 1, m m 1, q 1, m m 1, q 1 b, ; B, 1 d,, 1 d, ; L, 1 f, F, 1 f, F ; S 1, ;, 1, ;, 1,, 1, ;, 1, 0, 1 : m, : m, 1, 1 1, 1, 1, 1, 1, q 1:, q :, q q m m q m m q y a A c K c e E R e E 1, 1 1, 1, 1, 1, The series reresetatio of the -fuctio of two ariables defied ad rereseted by (.1) is as follows: (i) d d m m m;, m 1,,..., m ;, 0,1,,... For (ii) F f F f k k k;, k 1,,... m ;, 0,1,,... For : c, ; K, c, : e, E ; R, e, E :,,, ; :,,, ; 0,0: m, : m, 1, 1, 1, 1, 0,0:, q:, q y d d L f F f F S 1, m m1, q 1, m m 1, q (.1) (.) (.) Coyright to IJIRSET

6 ISSN: (A ISO 97: 007 Certified Orgaizatio) Vol., Issue 9, Setember 01 (iii), y 0, U 0, V 0 [U ad V are defied by (.5) ad (.6)], ad if all the oles of (.1) are simle, the the itegral (.1) ca be ealuated as a sum of residues to gie m ( 1) ( ) ( y) F!! m,,, 1 m,,, 4 m, 5, (.4) y m m d f, (.5) F m m,, m ( ) 4 m d 1 c 1 1 m q c 1 d 1 m1 5 is defied aalogously to 4 i terms of the arameter sets e, E ; R, e, E, f, F, f, F; S K 1, 1, 1, m m 1, q L Coyright to IJIRSET (.6). Also 1 is defied by meas of (.). By comarig the series reresetatio of [, y] ad -fuctio, we get the followig iterestig secial case of (.1). a, ; A, c, ; K, c,, e, E ; R, e, E, ;,,,, ;,,,, ; 0, 1 : m, : m, 1, 1 1, 1, 1, 1, 1, q 1:, q :, q y b B d d L f F f F S 1, q1 1, m m 1, q 1, m m 1, q w0 w w ( 1) y 1 e E 1 q w w F w! e E 1 f F 1 1 w c, ; K, a Aw,, c, d,, d, ; L, b w, B m, 1 1, 1, 1 1, q, q1 q 1, m m 1, q 1, q1 R I (.7), whe we ut f 0, F 1ad y 0, we get the followig useful cofluet of the -fuctio of two ariables: lim y0 S a, ; A, c, ; K, c,, e, E ; R, e, E, ;,,,, ;,(0,1),, ; 0, 1 : m, :1, 1, 1 1, 1, 1, 1, 1, q1:, q:, q 1 y b B d d L f F S 1, q1 1, m m1, q 1, q (.7)

7 ISSN: (A ISO 97: 007 Certified Orgaizatio) 1e 1 q F w! e 1 f 1 1 R Vol., Issue 9, Setember 01 S a, ; A, c, ; K, c, b, ; B, d,, d, ; L m, 1 1, 1 1, 1, q, q1 q 1, q1 1, m m1, q Also makig use of a result [, ol.1,.104, eq. (46)] ad (.1) i (.7), we obtai aother iterestig result: a, ;1,( b, ;1), c, ; K, c,, d d L f h 0,1: m, :1,0 1, 1,,0:, q:0, 1,,,, ;,,1,,1;1 1, m m1, q m 1, 1 1, 1, 4, q1 ( ab f h1, ), d,, d, ; 1, m L m 1, q Re[ a b f h 1] 0, 0, 0. The fuctios a f, ;1, c, ; K,, c,,( b f, ),( bh, ),( ah, ) J y ad q y IV. SPECIAL CASES (.8) (.9) are the Wright s geeralized Bessel fuctios of two ariables ad geeralized hyergeometric fuctio of two ariables resectiely. The study of the -fuctio of two ariables will coer wider rage the the -fuctio of two ariables ad gies deeer, more geeral ad more useful results directly alicable i arious roblems of hysical ad biological scieces ,0:1,0:1,0 : : ' 0,0:0,:0, y :(0,1),(, ; K ),(0,1),( ', '; S) y ' y J J : 1 c, ; K, 1 e, E ; R :(0,1), 1, ;,(0,1), 1, ; 0,0:1, :1, 1, 1, 0,0:, q 1:, q 1 y d L f F S 1, q 1, q c, ; K 1, e, E ; R 1, y d, ; L f, ; 1, q F S 1, q q q ' 0,0:1,0:1,0 a a 4 : : y 0,0:0,:0, J ( ) ' ' ' ' J ' ( y) y a a a a :,1,,1; L,,1,,1; 4 S (.) J ad (.) q hyergeometric fuctio resectiely ad defied as [4]: (4.1) (4.) (4.) are kow as Wright s geeralized Bessel fuctio ad Wright s geeralized r 1,0 0, ( ) ( ) (0,1),(, A J ; ) (4.4) A r0 r! 1 r Coyright to IJIRSET 490

8 ISSN: (A ISO 97: 007 Certified Orgaizatio) Vol., Issue 9, Setember 01 Ad 1, 1 a, ; A, ; 1, a A 1, q, 1 (0,1), 1 b, ; B q b, ; 1, q B 1, q A a r r 1 (4.5) q B r0 r! b r 1 REFERENCES [1]. Buschma, R.G. & Sriastaa,.M.; The -fuctio associated with a certai class of Feyma Itegrals. J. Phys. A. Math. Ge.: ,1990. []. Erdelyi,A., Magus, W., Oberhettiger, F ad Tricomi, F.G.; igher Trascedetal Fuctios. Vol. I, McGraw-ill, NewYork, 195. []. Fo, C. The G ad -fuctio as Fourier kerels. Tras. Amer. Math. Soc.98:, 95-49, [4]. Guta,K.C., Jai, R. ad Sharma, A.; A study of uified fiite itegrals with alicatios. J. Raastha Acad. Phys. Sci. Vol.(4):,69-8, 00. [5]. Mittal, P.K. & Guta, K.C.; A itegral iolig geeralized fuctio of two ariables. Proc. Idia Acad. Sci. Sect. A( 75):, 67-7, [6]. Rathi, A.K. A ew geeralizatio of geeralized hyergeometric fuctios. Le Mathematiche Fasc. II5:, 97-10, [7]. Sriastaa,.M., Guta, K.C. & Goyal,S.P.; The -fuctios of Oe ad Two Variables With Alicatios. South Asia Publishers, New Delhi,198. Coyright to IJIRSET 491