THE MELLIN-BARNES TYPE CONTOUR INTEGRAL REPRESENTATION OF A NEW MITTAG-LEFFLER TYPE E-FUNCTION

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1 AMRICAN JOURNA OF MATHMATICA SCINC AND APPICATIONS 2(2) Jly-December 24 ISSN : X pp TH MIN-BARNS TYP CONTOUR INTGRA RPRSNTATION OF A NW MITTAG-FFR TYP -FUNCTION Sy Btter d Se Mommed Fsl Deprtmet of Mtemtcs, Mly Ntol Isttte of Tecology, Jpr, Rst-327, Id -ml: [email protected] Abstrct: I ts pper, we stdy ew Mttg-effler type -fcto tt fes my specl fctos d dere ts Mell-Bres type cotor tegrl represetto. Keywords: Mttg-effler fctos. 2 MSC: Prmry 332; Secodry 332. INTRODUCTION Te Mttg-effler (M-) fcto trodced 93 de to Göst Mttg-effler s geerlto of te expoetl fcto e. Sce te orey of M- fcto strted s geerlto of expoetl fcto d lter ts my geerltos were stded by my tors, mottes te resercer to stdy ew geerlto of Mttg-effler type fcto. 2. INTRODUCTION I 24, Btter d Fsl [], defed fed M- type -fcto s follows ( );(, q, s ), (, );(, q, s),...,(, q, s) (, );(, p, r ), (, );(, p, r ),...,(, p, r ) s s2 s ( ) q ( 2) q... ( ) ( ) 2 q r2 r ( ) ( )... ( ) ( ) p 2 p2 p () were or,,,, ; ( ), ( ), ( ), ( ), ( q ), ( p ) ; s, r,, ; {,}, qs p r ( ) p r q s p r ( ) we ( q ) ( p ) qs for =, 2,..., ; =, 2,...,. (2)

2 38 SANJAY BHATTR AND SHIKH MOHAMMD FAISA 2. TH MIN-BARNS TYP CONTOUR INTGRA RP-RSNTATION Teorem 2.: et coergece codtos (2) re stsfed te te -fcto [ ] c be represeted s te Mell-Bres type cotor tegrl s follows r )] (, ); (, q, s),...,(, q, s ) ( ) (, );(, pr ),...,(, p, r ) s 2 )] [( ) ] ) ] q )] ( ) p )] r s ( ) d were s stble cotor of tegrto tt rs from c to c +, c d teded to seprte te poles of te tegrd t for ll (to te left) from tose t d t,, 2,..., ; for ll (to te rgt). Proof: Te -fcto c be wrtte s (, ); (, q, s ),..., (, q, s) (, ); (, p, r ),..., (, p, r ) r ( ) s )] ( ) q)] s r )] ( ) p )] ( ) (3) ( ) )] )] s r lm [( ) ] ) ] ( ) g( ) ( ) (4) were te s q)] g( ) ( ) p )]. H. S. )] )] r s ( ) 2 r (5)

3 TH MIN-BARNS TYP CONTOUR INTGRA RPRSNTATION [( ) ] [ ( ) ] q )] ( ) p )] r s ( ) d (6) Corollry 2.2: If coergece codtos (2) re stsfed te te -fcto [ ] c be represeted s te Mell-Bres type cotor tegrl s follows r )] (, );(, q, s ),...,(, q, s ) (, );(, p, r ),...,(, p, r ) s 2 )] ( ) ( ) q )] ( ) p )] s r [( ) ( )] d (7) were s stble cotor of tegrto tt rs from c to c +, c d teded to seprte te poles of te tegrd t = for ll (to te left) from tose t = + d t, q =, 2,..., ; for ll (to te rgt). 2.2 Specl Cses. For =, s =, =, r =, =, = ; = ; =, we get M- fcto () defed 93 by Göst Mttg-effler [4] s (,);(, q,) (,);(, p,) ( ) 2 ( ) ( ) ( ) d ( ). ( ) (8) 2. For =, s = ; =, r = ; = ; = ; =, we get geerled M- fcto () defed 95 by Wm [6] s (,);(, q,) (, );(, p,) ( ) 2 ( ) ( ) ( ) d, ( ). ( ) (9)

4 4 SANJAY BHATTR AND SHIKH MOHAMMD FAISA 3. For =, s = ; = m, r =... = r m =, = µ,..., m = m, p,..., p m ; ; ; ; ; m we get, ( ( ) ), ( ) defed 2 by Kryo [3] s m m m (,); (, q,) (, m ); (,,),..., ( m, m,) m m )] ( )... ( ) m m m )] ( ) ( ) m ( ) d )] ( ), ( ) ( ). m 2 () )] 4. For =, s =, =, g = ; =, r = =, p = ; = ; = ; =, we get M- type fcto,, ( ) defed 29 by Srst d Tomos 5 s (,);(,,) ( ) (, );(,,) ( )! ( ) ( ) () ( ), d, ( ). ( )2 ( ) 5. For =, s = ; =, r =... = r =, = + µ,..., = + µ, p =,..., p = ; ; ; M ; ; d replce by, we get,..., H,..., ( ) defed 22 by Kll, Hdey d Vrceo [2] s M (, );(, q,) (, );(,,),...,(,,) ( ) )] ( ) M d M (2 ) ( 2 ) )] )] )] H ( ).,...,,..., (2)

5 TH MIN-BARNS TYP CONTOUR INTGRA RPRSNTATION... 4 RFRNCS [] Btter, S., Fsl, S.M.; A Fmly of Mttg-effler Type Fctos d ts Relto wt Bsc Specl Fctos. It. J. of Pre Appl. Mt., (ccepted). [2] Kll, S.., Hdey, V. d Vrceo, N.O.; A Geerled Mltprmeter Fcto of Mttg-effler Type. Itegrl Trsforms Spec Fct, 23(2), 22, 9-9. [3] Kryo, V.S.; Mltple (mltdex) Mttg-effler Fctos d Reltos to Geerled Frctol Clcls. Jor. Comp. Appl. Mt., 8(-2), (2), [4] Mttg-effler, G.M.; Sr l Noelle Focto (x). CR Acd. Sc. Prs, 37, (93), [5] Srst, H.M. d Tomos, Z.; Frctol Clcls wt Itegrl Opertor Cotg Geerled Mttg-effler Fcto te Kerel. Appl Mt Compt., 2, (29), [6] Wm, A.; Über de Fdmetl St der Teore der Ftoe (x). Act Mt. 29, (95), 9-2.