Contour integral involving Fox's H-function and modified Bessel function of the first kind has been calculated. Some important properties and particular cases of H-function, which is a generalization of G-function, have been derived and discussed

Bajpai, S. D.

English

Defence Science Journal

SOME CONTOUR INTEGRALS INVOLVING GENERALIZED HYPERCEOXETRIC FUNCTION - 8. D. BAJPAI Shri G. S. Teohnological Institute, Indore* (Received 25 March 1969) - .  Contour integral involving Fox's H-Function and modified Bessel function of the &st kind has ,been calculated. Some important properties and particdar. oases of H-function, which is s generalization of G-function, have been derived and discussed. - 9 - = In  this paper we have evaluated a contour integralinvolving Fox's H-function and " modified Bessel function of the first kind. On specializing the parameters the integral yields many results scattered throughout the literature, some of which are given by MacRobertl and Meijer2. - - The H-function introduced by  FOX^ will be represented and defined as follows : where an empty product is interpreted as I, 0 < m 4 q, 0 4 lt $ p; e's andfa are all positive; L is a suitable contour of Barnes type such that the poles of P(bj  -fjs),j = 1, , . . . . . , m lie on the right hand side of the captour and those of r ( 1 - aj 4- ej s ), J 7 1 , . . . . . . , n lie on the left hand side of the contour. - Recently Braaksma4 has discussed, asymptotic expansions and analytic continua- tions for the H-function. Now we discuss some important properties and particular cases of the H-function, which are apparent from the definition of the H-function. The H-function is symmetric in pairs ( a, , el ) , . . . . . . . . . . . . , ( an , en ) likewise in (an+,,e,.+l), ............ , ( a p , e p ) ;  in (biJ f l ) ,  ........ , (bm, f m )  and in (brn+l,frn+l)., ........ , (bq , J? )*  Inone o f ( a j ,  ej) ( j  =I, .. . ..., n ) i s  samea~oneof(bj,fj)(j=m+l,.....,q)or oneof(bj ,  f j )  ( j = 1 ,  ...... ,m)issameasoneof(aj,ej)(g=lt+l,....-.,p)then the H-function reduces to one of lower order, i. e., eaoh of p, q qnd or dmreases by unity. - i * Present address :-Regional Engineering College, Kurukshetra. 111 and ( ( p  .)l eep ) represents the set of parameters ( 4 , e, ) , . . . . . . . . , ( @p t 61, ) a  I (2) (3) am' " [ z  1 "1 = G"' " [ Z  1: 1, P.)l!l i b  1 )  P , ¶  (4) where'the right hand side is Meijsr's G-function. , (5 )  where the right hand side is MaoRobert's E-functibn. % P r ! (6) ,=o J (bj+fjp \ j= 1 - -  . The above sex+es was staxdied by Wright5 and has been called as ~ r i i h t ' s  generalized 4 hypergeometric function and is denoted by the symbol [ !if:;:j ; - X I  ( A - k + I , I )  I X "P (1/2+A+td, 1 1 ,  (112 f A - / A )  1 )  = 2 e W k , p ( z ) ,  (7) where Wk, 1.c (t ) is a Whittaker function. 130 . rn ( - z I r  %A [ Z  / ( o J ) 2 ( - v > p ) ]  == 2 p ! , q l + v + p p ~  = J;(Z) ,  6-- 3 -0 ($1 . where J ( t ) is . MaitIand'a generalized Bessel funotion6. v 4 8 i : i [ z 2 / 4  l ( h / 2 - - ~ ! 2 , 1 ) ,  ( 4 2 + v / % , l )  (9) C where K K ~  ( z ) is a modified Bessel function. where h is a positive number, \" m. On applying (I), the formula (11 In (ll), putting v = - p , I-& (xy ) - where h is a positive number, < A G O ,  B > O ,  jargzl  < & B * ,  Re [ p + 2 h ( 1 - a j ) l e j ] >  O(j= 1 ,  ........, n ) .  0 P A R T I C U L A R  C A S E S  The H-function is a generalization of the G-function, which itself is a generalization of many higher transcendental functions8. Therefore, the formulae (11) and (12) are of general character and may ehcompass several cases of interest. It is interesting to note that the r e d &  (11) and (12) lead to generalization of many results of inverse K-transforms7 and inverse Laplace transforms9. However, some interesting particular cases are given below : (i) In (11) , subetituting e j  = fi = I ( Q = 1, . . . . . . , n ; i = 1, . . . . , m ) , assuming h as' a positive integer, using (4) and simplifying with the helpQf (I), we obtain where h i s  a positiveinteger, 2 (rn + n )  > p +  q ,  I argzl < (rn + n - - 4 p - - + q )  a ,  Re [ p + 2h ( 1 - ctj ) I > 1 Re v 1 - 4 ( j = 1 , . . . . . . . . , n ) and the symbol ( h , a ) represents the set of parameters cc a + l  a + h - 1  $ ,  , . . . . . . . . , - h In (13), taking h = 1, p = 0 and adjusting the parameters suitably, i t  reduces to a * known result7. (ii) In (12), replacing 2h by h, reducing the H-function to the G-function as above, using duplication formula for the gamma-functions and simplifying. we get -v $21 o - i w  where h is a positive integer, 2 ( ~ + n ) > ~ + q , l a r g z i ~ ( ~ + n - f ~ - 4 9 ) . 1 1 ,  * R e [ p + %  ( 1 - q ) ] > O ( j =  1 ,  ...... , N ) ,  r I n ( l 4 ) , s e t t i n g h = l , x = - 1 , p = l - ~ ,  m = l , b , = O , ~ = p ,  replacing y by i - y , q b y q i -  1, b k b ~ b k - ~ ( k  = 2 , 3 ,  ... ... ,p+l),andusingreferencelO, via., and deforming the contour suitably, i t  reduces to a result given by Meijer2. , . \  , (iii) I p  ( l l) ,  adjusting the suitably in view of (5) and simplifying, we get c + i w  112-P 25j ?I i v t x ? l ) E  c-im / In (15), putting h = l  and adjusting pardmeters, we get a known result'. (iv) In (12), replacing 2h by h, reducing the H-function to the E-function and eimplifying, we have -P zy ' /  y B ~ [ : : : ~ ~ ~ ] d ~  2Gi c-iob h 1 2 - l l 2  1P-P P - 1  [ czp  : z (hlx)% j = (2.I) la 2 b q , A ( h , P )  , (16) where h is a positive integer, p + 1 > g ,  1 arg z 1 < ( p - 9 + 1 ) , Re p > 0. In  (16), putting x = l  and deforming the contour suitably, we obtain a result given by MacRobertl. (4 In (11), replacing m  by 1, n by P, g byg  + 1, uj by 1 - y ( j = I ,  . . . . , p ), bl by O,fi by 1, bj by 1 - bj.. .~, f j  by fjl ( j = 2,3, . . . . , q  ) and using (6), we get o+iw where the conditions are corresponding to (11). (vi) In (ll),  taking m = q =  2 , n =  O , p =  l , a l = A - k + l , b l = ~ + ~ + p ,  b , = A - t ~ + $ ,  a n d q = f , = f i = l  and using (7), we get 116 . ~ E P .  BCI. S., VOL. 20, APRIL 1970 2h f 112-P ZY 12 Wk, P b y a h )  dy c-im --A P-312 2 ,  0  = 2  ( 2 )  H (A--k+ 1 , 1 ) ,  ( ( p + v ) / 2 +  1/4 ,h)  3 , s  (( p -V ) J2 3- 114, h 1 (4-t.A+~, I ) , ( h - p + Q ,  1) where the conditions are corresponding to (11). a ( v i i )  In (ll), setting m = 1, n = p = 0, q = 2, b, 0, b, F - u, fl = 1, fz = p and using (S), we obtain c f  im C-im - 2 h  i ( ~ + ~ ) / 2 + 1 / 4 , f i ) > i ( ~ - ~ ) l 2 + 1 / 4 ~ ~ ) ]  (19) = ( i 2 ) p 3 ' P ~ 1 3 0 [ z ( x i 2 )  / 2 , 2  l ( o , l ) , (  , where the conditions are corresponding to (ll). '  (9%) Puttingm=q=2,rz=p=O,bl=~X-+u,b,=+hf~~andcfi-~=1,the integral (ll) ,  in view of (9), reduces to cf ica 1/2-,7 L j  Y + b ~ ) K . ( 2 d ~ \ / ; ~ ) r l ? l  / 2?ri C-$00 where the co~lditions are corresponding to (11). A C K N O W L E D G E M E N T  I am thankful to Principal: Dr. S. M, Dae Gupta for the facilities he provided to me. i R E F E B E N C E S  1. XACWOBIPRT, 'l!. M., Proc.f#tasgow B d h .  Assoc., 1 (1953), 190. 2. M'EIJICB, C. S., Nedrel. Akad. Wetemch Proc., Ser. A, 56 (1963), 349. 3. Fox,C., Trana. Arner. Math. iSoc.,98 (1961), 395. 4. BRAAISMA, h. L. J., c0mpJ8. blath., 15 (1963), 239. 6. Wmam, E. N.., J .  London Mdh.  A%., 38 (1983), 286. 6. WRI~ET, E. M., Proc. &mdon Alalh. ~SOC., 10 (1936), 257. ,. 7. EEDELM, A., Tables of Integral Transfopme., Vol. 2 (McGraw-=ll, Wew York), 1954. 8. ERDPILYI, A., Higher Transcendental Functions, Vol.  1 (MoGraw-Bill, New York), 1963. 9. EEDELYI, k, Tables of Integral Transfom, Vol, 1 (McGraw-Hill, Now York), 1964. 10. MEIJEB, C. S., Nedrel. Aliad. WePate#ec;h. Pmc. Ser. A, 65 (1952), 369. 

Some Contour Integrals Involving Generalised Hypergeometric Function

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Some Contour Integrals Involving Generalised Hypergeometric Function

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