Saigo-Maeda Fractional Differential Operators of the Multivariable H-Function

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1 Intenatonal Jounal of Computatonal Scence and athematcs. ISSN Volume 5, Numbe (0, pp Intenatonal Reseach ublcaton House Sago-aeda Factonal Dffeental Opeatos of the ultvaable H-Functon Kantesh Gupta and eena Kuma Gua Depatment of athematcs, alavya Natonal Insttute of Technology, Japu-007, Raasthan (Inda E-mal: Abstact In ths pape, we study and develop the genealzed factonal dffeental opeatos nvolvng Appell s functon F (. [, p. 4, Eq (8] ntoduced by Sago and aeda [, p. 9] to the multvaable H-functon. Fst, we establsh two theoems that gve the mages of the multvaable H- functon n Sago-aeda opeatos. On account of geneal natue of these opeatos, a lage numbe of new and known theoems nvolvng Sago, Remann-Louvlle, Edély- Kobe factonal dffeental opeatos and seveal specal functons notably genealzed wght hypegeometc functon, ttag-leffle functon, genealzed laucella functon, Bessel functons follow as specal cases of ou man fndngs. The mpotant esults obtaned by Gupta [], Klbas [4], Klbas and Sago [5], Klbas and Sebastan [6], Saena, Ram and Sutha [7] and Saena and Sago [8] follow as specal cases of ou esults. Keywods: Sago-aeda factonal dffeental opeatos, Appell functon, Gauss hypegeometc functon, ultvaable H-functon, Bessel functon, ttag-leffle functon. AS Subect Classfcaton: 6A, C05, C0, C60, C65, E.. Intoducton: The factonal dffeental opeato nvolvng vaous specal functons, have been found sgnfcant mpotance and applcatons n vaous sub-feld of applcaton mathematcal analyss. Snce last fve decades, a numbe of wokes lke Kyakova [9], Svastava et al. [0], Saena et al. [, ], Sago [], Klbas [4], Klbas and

2 44 Kantesh Gupta and eena Kuma Gua Sebastan [6], Samko et al. [4], lle and Ross [5], and Gupta et al.[] etc. have studed n depth, the popetes, applcatons and dffeent etensons of vaous hypegeometc opeatos of factonal dffeentaton. Recently, Kuma and Daya [6] have appled ths factonal dffeentaton of the poduct of a geneal class of olynomal and _ H -functon. Saena, Ram and Kuma [] have also deved the genealzed factonal dffeentaton of the Aleph functon assocated wth the Appell functon F. Let αα,, ββ,, γ C, γ > 0 and > 0, then the genealzed factonal dffeentaton opeatos [] nvolvng Appell functon F as a kenel ae defned by the followng equatons: α, α, β, β, γ,,,, D f α α β β γ I f n d α, α, β + n, β, γ + n I f d, R ( γ > 0; n R γ + d n t ( α ( t n γ t α F α, α, β + n, β, n γ;, f ( t dt n γ d Γ 0 t ( and α, α, β, β, γ,,,, D f ( α α β β γ I f n d α, α, β, β + n, γ + n I f, R ( γ > 0; d n R γ + n d t ( α ( t n γ t α F α, α, β, n β, n γ;, f ( t dt n γ d Γ t (,,,, I αα β β γ,,,, and I αα β β γ ae Sago- aeda factonal ntegal opeatos. F the Appell-hypegeometc functon of two vaables s defned as ( α m n (,,, ; ;, m α n β m β n z ξ F αα ββ γ z ξ, ( z <, ξ < ( 0 0 ( γ m! n! m n m+ n ( z m and ( z n ae the ochhamme symbol defned by z C mn, N0 N { 0 }, N {,,,... } by ( z, ( z 0 m z( z+...( z + m. The above sees ( s absolutely convegent fo ( z <, ξ < and ( z, ξ ( z, ξ. and, These opeatos educe to Sago factonal devatve opeatos [; see also 7] as 0, α, β, β, γ,, D f ( γ α γ β γ D f, ( R ( γ > 0 ; (4

3 Sago-aeda Factonal Dffeental Opeatos of the ultvaable H-Functon 45 and 0, α, β, β, γ,, D f γ α γ β γ D f ( R ( γ >, 0. Futhe, we also have [, p. 94, Eqns. (4.8 and (4.9] αα ββ γ σ σσ, + γ α α βσ, + β α σ ( I t Γ σ + γ α α, σ + γ α β, σ + β α α γ,,,, + R ( γ > 0, R ( σ > ma 0, R ( α + α + β γ, R( α β σ Γ and αα,, ββ,, γ + α+ α γ σ, + α+ β γ σ, β σ σ α α + γ I t σ, + α+ α + β γ σ,+ α β σ (5 (6 (7 R ( γ > 0, R ( σ < + mn R( β, R ( α + α γ, R ( α + β γ. Hee, we have used the symbol L Γ epesentng the facton of many Gamma L functons. In ths pape, the multvaable H-functon wll be defned and epesented n the followng manne [0, p. 5-5, Eqs. (C.- C.]: 0, N :, N ;...;, N H[ z,..., z ] H, Q:, Q ;...;, Q ;,, ( :, ; ; (, ( a α L α c γ L c γ, ;,, :, ; ;, b β L β d δ L d δ z,, z, Q, Q, Q ξ (,, L ψ ξ L ξ z d d Π θ ξ ξ L ξ ( πω L L (8 ω ( N ( ( ( ( Γ( d - δ ξ Γ (- c + γ ξ θ ( ξ,,,..., Q ( ( ( ( Γ ( - d + δ ξ Γ( c - γ ξ + N + { } (9 and N ( Γ( a + α ξ Ψ ( ξ,, ξ ( Q ( Γ(a α ξ Γ( b + β ξ N + L (0

4 46 Kantesh Gupta and eena Kuma Gua In equaton (9, n the supescpt ( stands fo the numbe of pmes, e.g., b b, b b and so on; and an empty poduct s ntepeted as unty. ( Fo convenence, we use the notaton ; α,..., α fo membe aay ( ( ( a ; α,..., α,..., a ; α,..., α ( a, (,, whle γ stands fo ( c,,..., c, c, γ γ, (,,..., and so on. The natue of contous L, L, n (8, the vaous specal cases and othe detals of the functon ae gven n the book efeed above. Also, t s assumed thoughout the pesent wok that the vaous multvaable H- functon occung heen always satsfy the condtons of the estence coespondng appopately to those mentoned n the efeence gven above. L. elmnay Results: Lemma. Let αα,, ββ,, γ C be such that ( γ 0 R ( σ > mn.[0, R ( α + α + β γ, R( α β ] Then thee holds the elaton σ 0 Γ( σ Γ( σ γ + α+ α + β Γ( σ β+ α ( σ γ α β ( σ γ α α ( σ β αα,, ββ,, γ σ γ+ α+ α D t + Γ + + Γ + + Γ Lemma. Let αα,, ββ,, γ C be such that R ( γ > 0, R ( σ < + mn.[ R( β, R α β + γ, R α α + γ n ] ;( n [ R γ ] + R >, ( Then thee holds the elaton σ ( α β γ σ ( β σ ( α α γ σ Γ( σ Γ( α α β+ γ σ Γ( α + β σ αα,, ββ,, γ Γ + Γ + Γ + σ γ+ α+ α D t (. an Results We establsh mage fomulae fo the multvaable H-functon nvolvng Sago and aeda factonal dffeental opeatos ( and ( n tems of the multvaable H- functon. Theoem.. Let αα,, ββ,, γσ,, z C, > 0, ρ > 0, {,,..., } be such that R ( γ > 0, ( d R σ + ρ mn R mn 0, ( >, R α + α + β γ R α β δ

5 Sago-aeda Factonal Dffeental Opeatos of the ultvaable H-Functon 47 and ag z < Ω π, Ω > 0 Q ( N ( ( ( Q Ω α 0,,,..., β + γ N γ N + δ δ > { } then the followng fomula holds ρ ;,, ( :, ; ; (, ( zt a α α c γ L L c γ 0, :, ;...;,,, N N N, αα,, ββ,, γ σ D t H Q, :, Q ;...;, Q ( ( ( ρ zt ( b ; β,, β :( d, δ L ; L; ( d, δ, Q, Q, Q ρ z 0, N+ :, N ;...;, N σ γ + α + α A : C H +,Q+ :, Q ;...;, Q ρ B : D z ( A σρ ;,..., ρ, σ+ γ α α β; ρ,..., ρ, σ α+ βρ ;,..., ρ, a ; α,..., α ( ( B σ+ γ α β; ρ,..., ρ, σ+ γ α α; ρ,..., ρ, σ+ β; ρ,..., ρ, b ; β,..., β ( ( ( ( (, ; ;,,, ; ;, ( γ L γ δ L δ ( (,, Q C c c D d d,,, Q, Q oof: In ode to pove (, we fst epess the multvaable H-functon occung n the left hand sde of ( n tems of elln-banes contou ntegal wth the help of equaton (8 and ntechangng the ode of ntegaton, we obtan (say I ξ...,,,,... (,..., σ ρξ ρ ξ αα ββ γ I Ψξ ξ z D t d... d θ ξ ξ ξ 0+ ( πω L L ( at Now, applyng the equaton ( wth σ eplaced by σ + ρξ ρ ξ, we ave (... (... ( σ γ α β ρξ ρξ ( σ β ρξ ρξ ξ... (,..., Γ σ+ ρξ + + ρ ξ Γ σ γ + α+ α + β + ρξ + + ρ ξ I Ψξ ξ θ ξ z ( πω L L Γ Γ ( ( σ γ α α ρ ξ... ρξ Γ σ β + α + ρ ξ ρ ξ σ γ+ α+ α + ρ ξ ρξ dξ... dξ Γ

6 48 Kantesh Gupta and eena Kuma Gua Fnally, ntepetng the elln-banes contou ntegal thus obtaned n tems of the multvaable H-functon as gven n (8, we obtan the esult as gven n (. 4. Specal Cases of Theoem.:. If we put α α + β, α β 0, β η, γ α n Theoem., we get known esult obtaned by Gupta et al. [, p. 48, Eq. (8], as ρ z t 0, N :, N ;...;, N α, β, η σ D t H,Q :, Q ;... ;, Q ρ z t ρ z 0, N+ :, N ;...;, N A : C σ + β H +,Q+ :, Q ;...;, Q ρ B : D z A σ ; ρ,..., ρ, σ α β η; ρ,..., ρ, a ; α,..., α B σ β; ρ,..., ρ, σ η; ρ,..., ρ, b ; β,..., β ( (,, Q C and D ae same as gven n ( and the condtons of estence of the above esult can be easly deved wth the help of Theoem... If we take N Q 0 n equaton (, the multvaable H-functon educes to the poduct of -dffeent Fo H-functons [8, p. (v, Eq. (A.4] followng as:,,,,, N ρ αα ββ γ σ D t H z t, Q ρ z 0, :, N ;...;, N A : C σ γ + α + α H, :, Q ;... ;, Q ρ B : D z ( σ ; ρ,.., ρ,( σ γ α α β ; ρ,.., ρ,( σ α β; ρ,.., ρ ( σ γ α β ; ρ,.., ρ,( σ γ α α ; ρ,.., ρ,( σ β; ρ,.., ρ A + + B C and D ae same as mentoned n equaton ( and the condtons of valdty of the above esult can be easly deved wth the help of Theoem.. (4 (5

7 Sago-aeda Factonal Dffeental Opeatos of the ultvaable H-Functon 49. If we educe poduct of - dffeent H-functons occung n the left hand sde of (5, to the - poduct of Bessel functon of fst knd [0, p.8, Eq. (.6.5], we get,,,, ρ α α β β γ σ D t J a t υ ρ a ρ υ σ γ + α + α 0, :, 0;...;, 0 A : (6 a H, : 0,;...; 0, B : D ρ a A σ ;,...,, ;,...,, ;,..., ρ υ ρ ρ σ γ α α β ρ υ ρ ρ + σ α + β ρ υ ρ ρ B σ γ α β ;,...,, ;,...,, ;,..., ρ υ ρ ρ σ γ α α ρ υ ρ ρ + + σ + β ρ υ ρ ρ D ( 0,,( υ,;...;( 0,,( υ, v. Now, educng the H-functon of seveal vaables occung n the ght hand sde of (6 to genealzed Laucella functon [9], we get ρ a υ,,,, ρ αα ββ γ σ γ α α l m n D t σ + + Γ Γ Γ J a t 0 υ + Γ ( υ Γ p Γ q Γ s + ρ ρ ( l; ρ,..., ρ,( m; ρ,..., ρ,( n; ρ,...,ρ : 0;...;0 a a F,..., :;...; ( p; ρ,..., ρ,( q; ρ,..., ρ,( s; ρ,..., ρ : ( υ +, ;...;( υ +, ( l σ + ρ, m, n υ σ γ + α + α + β + ρ υ σ + α β + ρ υ p σ γ + α + β +, q, s ρ υ σ γ + α + α + ρ υ σ β + ρ υ v. If we take, a, ρ, υ υ n above equaton (7, we get σ + υ γ + α + α 0 υ α, α, β, β, γ { D ( t σ Jυ ( t } + (7

8 50 Kantesh Gupta and eena Kuma Gua ( σ + υ,,( σ + υ γ + α + α + β,,( σ + υ + α β, ( σ + υ γ + α + β ( σ + υ γ + α + α ( σ + υ β ( υ + ψ ; 4,,,,,,, 4 (8 Futhe, takng α α + β, α β 0, β η, γ α n above equaton (8, we get known esult due to Klbas and Sebastan [6, p. 0, Eq. (], as σ + υ + β α, β, η σ ( σ + υ,,( σ + υ + α + β + η, { D ( t J ( t } ; 0 υ υ ψ + ( σ + υ + β,,( σ + υ + η,,( υ +, 4 v. If we educe the multvaable H- functon nto the poduct of two Fo H-functon and then educe one H- functon to the eponental functon by takng ρ n equaton (, we get the followng esult afte a lttle smplfcaton: ( c,, γ,,,, zt, N ρ α α β β γ σ D t e H z t, Q ( d, δ, Q A 4 : ;( c, 0,:, 0;, γ z σ γ + α + α, H,: 0,;, ρ :( 0, ;(, δ N Q z B 4 d, Q ( ;,, ( ;,, ( ;, ( ;,,( ;,,( ;, A 4 σ ρ σ + γ α α β ρ σ α + β ρ B 4 σ + γ α β ρ σ + γ α α ρ σ + β ρ The condtons of valdty of the above esult can be deved wth the help of Theoem.. Futhe, on lettng z 0 n the above equaton (9, t becomes (, γ ( δ c,,,,,, N ρ α α β β γ D t σ H z t, Q d,, Q, σ γ α α N + ρ A + + H z 5, + Q + B 5 (, γ ;( σ, ρ,( σ γ α α β, ρ,( σ α β, ρ,(,, N γ N +, (, δ ;( σ γ α β, ρ,( σ γ α α, ρ,( σ β, ρ A 5 c + + c B 5 d, Q (9 (0

9 Sago-aeda Factonal Dffeental Opeatos of the ultvaable H-Functon 5 Now, we put α α + β, α β 0, β η, γ α n above equaton (0, we get known esult due to Gupta et al. [, p. 5, Eq. (4], as (, γ (, δ c,,, α β η, N ρ σ D t H z t, Q d, Q ( c,,(,,(,,(,, γ σ ρ σ α β η ρ c, γ N + σ + β, ρ N N + H z, + Q + ( d, δ,( σ β, ρ,( σ η, ρ, Q Agan, takng β α n the above equaton and make sutable adustment n the paametes, we get also known esult ecoded n the book by Klbas and Sago [5, p. 55, Eq. (.7.], as ( c,, γ, N ρ α σ D t H z t, Q ( d, δ, Q ( c, γ,( σ, ρ,( c, γ (, δ,( σ + α, ρ, N, N N, σ α + ρ + H z, + Q + d, Q v. If we educe the H- functon to genealzed Wght hypegeometc functon [0, p. 9, Eq. (.6.] n equaton (0 by takng, N N,, Q Q, we get ( c,,,,, γ αα ββ γ σ ρ, D t ψ z t Q ( d, δ, Q ( c, γ, σ, ρ, σ γ + α+ α + β,, ρ, σ+ α β, ρ σ γ + α+ α ρ ψ z + Q+ ( d, δ,( σ γ α β, ρ,( σ γ α α , ρ,( σ β, ρ, Q Now, we put α α + β, α β 0, β η, γ α n equaton (, we get known esult due to Gupta et al. [, p. 5, Eq. (5], as ( c,,, γ c,, γ,, σ, ρ, σ+ α+ β+ η, ρ αβησ ρ σ+ β ρ D t ψ z 0 t ψ z + Q ( d, Q δ + + ( d,,(,,(,, Q δ σ+ β ρ σ+ η ρ, Q Futhe, takng β α n the above esult, we get also known esult obtaned by Klbas [4, p. 9, Eq. (4], as ( c, γ c,, γ,, σ, ρ α σ ρ σ α ρ D t ψ z 0 t ψ z + Q ( d, Q δ + + ( d,,(,, Q δ σ α ρ, Q ( ( (

10 5 Kantesh Gupta and eena Kuma Gua v. If we put z, ρ n equaton (0 and educng the H-functon nto genealzed ttag-leffle functon [0], we get σ γ + α + α αα,, ββ,, γ { D t σ ρ E ( t } uv, Γ ( ρ, 4 ( ρ,,( σ,,( σ + γ α α β,,( σ α + β, H 4,5 (4 ( 0,, ( σ + γ α β,, ( σ + γ α α,, ( σ + β,, ( vu, Now, we put α α + β, α β 0, β η, γ α n equaton (4, we get known esult due to Gupta et al. [, p. 5, Eq. (7], as σ + β αβη,, σ ρ, ( ρ,,( σ,,( σ α β η, { D ( t E ( t } H uv,,4 Γ ρ ( 0,, ( σ β,, ( σ η,, ( vu, Futhe, takng β α n the above esult, we get also known esult due to Saena et al. [7, p. 70, Eq. (.6], as σ α α σ ρ, ( ρ,,( σ, { D ( t E ( t } H u, v, Γ ρ ( 0,,( σ + α,,( v, u. If we take N, Q, z a, ρ β, σ γ, c δ, γ, d 0, δ, d γ, δ β n equaton (, we get known esult due to Saena and Sago [8, p. 49, Eq. (9], as ( { α γ δ β } γ α D t E at δ β E 0,, ( a + β γ β γ α Theoem.. Let αα,, ββ,, γσ,, z C, > 0, ρ > 0, {,,..., } be such that R ( γ > 0, and d R σ ρ mn R < + mn R( β, R( α β + γ, R( α α + γ n δ ag z < Ω, 0 π Ω > Q ( N ( ( ( Q Ω α 0,,,..., β + γ N γ N + δ δ > { } then the followng fomula holds

11 Sago-aeda Factonal Dffeental Opeatos of the ultvaable H-Functon 5 ρ z t 0, N :, N ;...;, N α, α, β, β, γ σ D t H,Q :, ;...;, Q Q ρ z t ρ z 0, N+ :, N ;...;, N σ γ + α + α A : C H +,Q+ :, Q ;...;, Q ρ B : D z ( ( ( (5 A σ γ + α + β; ρ,..., ρ, σ β; ρ,..., ρ, σ γ + α+ α; ρ,..., ρ, a ; α,..., α, ( ( σ; ρ,..., ρ,( σ γ α α β; ρ,..., ρ,( σ α β ; ρ,..., ρ, b ; β,..., β B , Q C and D ae same as gven n (. oof: The poof of Theoem. can be developed on the lnes smla to those gven wth poof of Theoem. wth the help of equaton (. A numbe of seveal specal cases of Theoem. can also be obtaned smlaly. ( Refeences: [] Edély, A., agnus, W., Obehettnge, F. and Tcom, F.G., 95, Hghe Tanscendental Functons, Vol. I, c Gaw Hll Book Company, New Yok, Toonto and London. [] Sago,. and aeda, N., 996, oe Genealzaton of Factonal Calculus. Tansfom ethods and Specal Functons, Vana, Bulgaa, pp [] Gupta, K.C., Gupta, K. and Gupta, A., 0, Genealzed Factonal Dffeentaton of the ultvaable H-Functon, J. of the Appled athematcs, Statstcs and Infomatcs (JASI, 7(, pp [4] Klbas, A. A., 005, Factonal Calculus of the Genealzed Wght Functon, Fact. Calc. and Appled Anal., 8(, pp.-6. [5] Klbas, A. A. and Sago,., 004, H-Tansfoms Theoy and Applcatons, Chapman & Hall/ CRC London, New Yok. [6] Klbas, A. A. and Sebastan, N., 008, Genealzed Factonal Dffeentaton of Bessel Functon of the Fst Knd, athematca Balkanca,, pp.-46. [7] Saena, R. K., Ram, J. and Sutha, D. L., 009, Factonal Calculus of Genealzed ttag-leffle Functons, J. Indan Acad. ath., (, pp

12 54 Kantesh Gupta and eena Kuma Gua [8] Saena, R. K. and Sago,., 005, Cetan opetes of Factonal Calculus Opeatos Assocated wth Genealzed ttag- Leffle Functon, Fact. Calc. and Appled Anal., 8(, pp [9] Kyakova, V., 008, A Bef Stoy About the Opeatos of the Genealzed Factonal Calculus, Fact. Calc. Appl. Anal., (, pp [0] Svastava, H.., Gupta, K. C. and Goyal, S.., 98, The H- Functons of One and Two Vaables wth Applcatons, South Asan ublcatons, New Delh, adas. [] Saena, R. K., Ram, J. and Kuma, D., 0, Genealzed Factonal Dffeentaton fo Sago Opeatos Involvng Aleph-Functon, J. Indan Acad. ath., 4(, pp [] Saena, R. K., Ram, J. and Kuma, D., 0, Genealzed Factonal Dffeentaton of the Aleph-Functon Assocated wth the Appell Functon F, Acta Cenca Indca, 8(4, pp [] Sago,., 978, A Remak on Integal Opeatos Involvng the Gauss Hypegeometc Functons, ath. Rep. Kyushu Unv.,, pp [4] Samko, S. G., Klbas, A. A. and achev, O. I., 99, Factonal Integals and Devatves, Theoy and Applcatons, Godon and Beach Sc. ubl., New Yok. [5] lle, K. S., and Ross, B., 99, An Intoducton to Factonal Calculus and Factonal Dffeental Equatons, John Wley and Sons Inc., New Yok. [6] Kuma, D. and Daya, J., 0, Genealzed Factonal Dffeentaton of the _ H -Functon Involvng Geneal Class of olynomals, Int. J. ue Appl. Sc. Technol., 6(, pp [7] Saena, R. K. and Sago,., 00, Genealzed Factonal Calculus of the H- Functon Assocated wth Appell Functon F, J. Fact. Calc., 9, pp [8] ukheee, R., 004, A Study of Geneal Sequence of Functons, ultdmensonal Factonal Calculus Opeatos and the Geneal H- Functon of One o oe Vaables wth Applcatons, h.d. thess, Unv. of Raasthan, Japu, Inda. [9] Svastava, H.., Daoust,. C., 97, A Note on the Convegence of Kampe De Feet s Double Hypegeometc Sees, ath. Nach., 5, pp [0] abhaka, T. R., 97, A Sngula Integal Equaton wth a Genealzed ttag-leffle Functon n the Kenel, Yokohama ath. J., 9, pp.7-5.