Note On Euler Type Integrals

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1 Iteatioal Bulleti of Matheatical Reseach Volue 2, Issue 2, Jue 25 Pages -7, ISSN: Note O Eule Type Itegals Meha Chad ad Eauel Guaiglia 2 Depatet of Matheatics, Fateh College fo Woe, Bathida-53 (Idia eha.jalladha@gail.co 2 Depatet of Physics E. R. Caiaiello, Uivesity of Saleo, 8484 Fisciao (Italy eguaiglia@uisa.it Abstact I this pape we fist evaluate Six fiite uified itegals ivolvig poduct of the Geealized Mittag-Leffle fuctio E (γ j,(l j (ρ j,λ z,, z] ad the geealized polyoials S x].the values of the itegals ae obtaied i tes of ψ(z (the logaithic deivative of Γ(z. O accout of the geeal atue of ou ai itegals a lage ube of ew ad kow fiite itegals ad othe itegals ivolvig siple Special fuctios ad Polyoials, follow as its special cases. Fo the sake of illustatio, we ecod hee soe special cases of ou ai itegals which ae also ew ad of iteest by theselves. The itegals established hee ae basic i atue ad ae likely to fid useful applicatios i seveal fields of sciece ad egieeig. Itoductio ad defiitios I ecet yeas the iteest i fuctios of Mittag-Leffle type aog scietists, egiees ad applicatios-oieted atheaticias has deepeed. The Mittag-Leffle fuctio aises atually i the solutio of factioal ode itegal equatios o factioal ode diffeetial equatios, ad especially i the ivestigatios of the factioal geealizatio of the kietic equatio, ado walks, Levy flights, supe-diffusive taspot ad i the study of coplex systes. The odiay ad geealized Mittag-Leffle fuctios itepolate betwee a puely expoetial law ad powe-law like behavio of pheoea goveed by odiay kietic equatios ad thei factioal coutepats, see Lag 8, 9], Hilfe 7], Saxea et al.7]. I 93, the Swedish atheaticia Gosta Mittag-Leffle ] itoduced the fuctio E α (z, defied as E α (z z Γ(α +, (. whee z is a coplex vaiable ad Γ(. is a Gaa fuctio, α. The Mittag-Leffle fuctio is a diect geealizatio of the expoetial fuctio to which it educes fo α. Fo < α < it itepolates betwee the pue expoetial ad a hypegeoetic fuctio z. Its ipotace is ealized duig the last two decades due to its ivolveet i the pobles of physics, cheisty, biology, egieeig ad applied scieces. Mittag-Leffle fuctio atually occus as the solutio of factioal ode diffeetial equatio o factioal ode itegal equatios. The geealizatio of E α (z was studied by Wia 26] i 95 ad he defied the fuctio as E α,β (z z, (α, β C; R(α >, R(β >. (.2 Γ(α + β which is kow as Wia s fuctio o geealized Mittag-Leffle fuctio as E α, (z E α (z. The foe was itoduced by Mittag-Leffle] i coectio with his ethod of suatio of soe diveget seies. I his papes, ], he ivestigated cetai popeties of this fuctio. The fuctio defied by (.2 fist appeaed i the wok of Wia 26]. The fuctio (.2 is studied, aog othes, by Wia 26], Agawal ], Hubet 5] ad Hubet ad Agawal 6] ad othes. The ai popeties of these fuctios ae give i the book by Edelyi et al. (3], Sectio Received: Jue, 25 Keywods: Fiite itegals, Sivastava polyoials, Geealized Mittag-Leffle fuctio, logaithic deivative. AMS Subject Classificatio: 4C, 33C45, 26A33.

2 2 Meha Chad ad Eauel Guaiglia 8. ad a oe copehesive ad a detailed accout of Mittag-Leffle fuctios ae peseted i Dzhebashya (2], Chapte 2. I 97, Pabhaka 2] itoduced the fuctio E γ α,β (z i the fo of E γ α,β (z (γ z Γ(α + β!, (.3 whee α, β, γ C; R(α >, R(β >, R(γ > ad (λ deotes the failia Pochhae sybol o the shifted factoial, sice (! ( N (λ Γ(λ + Γ(λ { ( ; λ C {} λ(λ + (λ + ( N; λ C (.4 Recetly geealizatio of Mittag-Leffle fuctio E γ α,β (z of (.3 studied by Sivastava ad Toovski 23] is defied as follows: E γ,k α,β (z (γ K z Γ(α + β!, (.5 whee α, β, γ C; R(α >, R(β >, R(γ > ; R(K > which, i the special case whe K q(q (, N ad i{r(α, R(β} (.6 was cosideed ealie by Shukla ad Pajapati 25]. A ultivaiable aalogue of Mittag-Leffle fuctio defied i (.3 is vey ecetly studied by Gouta 4] ad Saxea et al. 6, p. 536, Eq..4] i the followig fo: E (γj (ρ j,λ z,, z ] E (γ,γ (ρ,,ρ,λ z,, z ] whee λ, γ j, ρ j C; R(ρ j > ; j, 2,,. k,,k (γ k (γ k z k z k Γ(λ + k ρ + + k ρ (k!(k!, (.7 If we take ρ ρ 2 ρ, the Equatio (.7 educes to the followig cofluet hypegeoetic seies 22, p.34, Eq. (.4(8]: Φ ( 2 γ,, γ ; λ; z,, z ] Γ(λ (γ i ki i (λ k+k k,,k whee λ, γ j, z j C (j, 2,, ad ax{ z,, z } < ; λ / Z. z k zk, (.8 (k!(k! A ild geealizatio of ultivaiable aalogue of Mittag-Leffle fuctio i (.7 is also due to Saxea et al. defied as follows 6, p.547, Eq. (7.]: (ρ j,λ z,, z ] k,,k (γ kl (γ kl z k z k Γ(λ + k ρ + + k ρ (k!(k!, (.9

3 Note O Eule Type Itegals 3 whee λ, γ j, ρ j C; R(ρ j > ; R(l j > ; l j N(j, 2,, ; λ / Z. Sivastava polyoials S x] will be defied ad epeseted as follows 9, p., Eq. (]: S x] /] l ( l A,l x l, (. l! whee,, 2,, is a abitay positive itege, the coefficiets A,l (, l ae abitay costats, eal o coplex. S x] yields ube of kow polyoials as its special cases. These iclude, aog othe, the Jacobi polyoials, the Bessel Polyoials, the Lague Polyoials, the Bafa Polyoials ad seveal othes (Sivastava ad Sigh, 983 2]. The followig esults ad defiitios ae also equied i ou ivestigatios. Pabhake ad Sua 3] defied the polyoials L (α,β (x as: whee α C +, β C + ad N. If α, the (. educes as: L (α,β Γ (α + β + (x Γ ( + L (,β Γ ( + β + (x Γ ( + whee L β (xis well-kow geealized Laguee polyoials (Raiville 4]. The Kohause polyoials of secod kid (Sivastava 2] is defied as: whee β C +, N ad k Z. It ca be easily veified that: Z β (x; k Γ (k + β +! k j k ( k x k Γ (αk + β +, (. ( k x k Γ (k + β + Lβ (x, (.2 ( ( j x kj j Γ (kj + β +, (.3 L (k,β ( x k Z β (x; k, (.4 The polyoials Z (α,β (x; k is defied24] as: Z (α,β (x; k Whee α C +, β C +, N ad k Z. Fo (.3 ad (.6, we get: L β (x Z β (x;, (.5 j If α N the (.6 ca be witte i the followig fo: Z (α,β Γ (k + β + (x; k Γ (α + Γ (k + β + ( j x kj j!γ (kj + β + Γ (α αj +, (.6 Z (,β (x; k Z β (x; k. (.7 ( α α x k. (.8 (α!γ (k + β + (

4 4 Meha Chad ad Eauel Guaiglia The set of polyoials L (α,β (γ; x is defied 24] as: whee α, γ C +, β C +, N. Fo (.9 ad (., we have: Oe ca easily veify that: L (α,β (γ; x Soe facts ae listed below (see Spaie ad Oldha 8] Γ (α + β + ( x!γ (α + β + Γ (γ γ +, (.9 L (α,β (; x L (α,β (x. (.2 L (k,β ( α; x k Z (α,β (x; k, (.2 Z (,β (x; L β (α; x, (.22 Z (,β (x; Z β (x; L β (x, (.23 L (,β (; x L (,β (x L β (x. (.24 ( x ( (x +, (.25 ( (x + y (x j j (y j, (.26 j (x + (x (x + ad (.27 ( x ( ( x. (.28! The followig well kow Eule itegal Foula is also equied to establish the ai itegals22, p. 275, Eq. 3]: u α u α ( u u β du du Γ(α Γ(α Γ(β Γ(α + + α + β, (.29 whee u j (j,,, u + + u ; R(α j >, j,, ; R(β >. 2 Mai Itegals Let ψ(z deotes the logaithic deivative of gaa fuctio Γ(z i.e. ψ(z Γ (z Γ(z, the we the followig esults. Theoe 2. If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u 2 x ( x x t S xλ ( x x η] log(x (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx /] ( k (γ kl A (γ kl z k z k,k Γ(λ + k ρ + + k ρ (k!(k! k k,k Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk i ( ] ψ(s + λ k + µ k ψ (s i + λ i k + µ i k i + δ i k i + t + ηk. i (2.

5 Note O Eule Type Itegals 5 Theoe 2.2 If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u 2 x ( x x t S xλ ( x x η] log(x (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx /] ( k (γ kl A (γ kl z k z k,k Γ(λ + k ρ + + k ρ (k!(k! k k,k Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk i ( ] ψ(s + λ k + µ k ψ (s i + λ i k + µ i k i + δ i k i + t + ηk. i (2.2 Theoe 2.3 If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u 2 x ( x x t S xλ ( x x η] log( x x (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx /] ( k (γ kl A (γ kl z k z k,k Γ(λ + k ρ + + k ρ (k!(k! k k,k Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk i ( ] ψ(t + ηk + δ k + + δ k ψ (s i + λ i k + µ i k i + δ i k i + t + ηk. i (2.3 Theoe 2.4 If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u 2 x ( x x t S xλ ( x x η] logx x ( x x ] (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx /] ( k (γ kl A (γ kl z k z k,k Γ(λ + k ρ + + k ρ (k!(k! k k,k Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk i ( ] ψ(s i + λ i k + µ i k i + ψ(t + ηk + δ k + + δ k ( + ψ (s i + λ i k + µ i k i + δ i k i + t + ηk. i i (2.4 Theoe 2.5 If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + +

6 6 Meha Chad ad Eauel Guaiglia δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u (x x log ( x x /] k k,,k xλ ( x x η] 2 x ( x x t S ] (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx ( k (γ kl (γ kl z k z k Γ(λ + k ρ + + k ρ (k!(k! Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk i ( ] ψ(s i + λ i k + µ i k i ψ(t + ηk + δ k + + δ k ( ψ (s i + λ i k + µ i k i + δ i k i + t + ηk. i Theoe 2.6 If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u ( x x log (x x /] k k,,k i xλ ( x x η] 2 x ( x x t S ] (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx ( k (γ kl (γ kl z k z k Γ(λ + k ρ + + k ρ (k!(k! Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk i ψ(t + ηk + δ k + + δ k ( ] ψ(s i + λ i k + µ i k i + ( ψ (s i + λ i k + µ i k i + δ i k i + t + ηk. i The followig iteestig itegal, which is established i equatio (2.7 will be equied to establish the esults fo (2. to (2.6. i (2.5 (2.6 Theoe 2.7 If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u 2 x ( x x t S (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx /] ( k (γ kl A (γ kl z k z k,k Γ(λ + k ρ + + k ρ (k!(k! k k,,k Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk /] k i xλ ( x x η] Γ(s + λ kγ(s + λ kγ(t + λk ( k E (γj,sj+λjk,t+ηk,(lj,µj, δ j (ρ j,(µ j+δ j,λ,t+ηk+ (s z j+λ jk,, z ]. (2.8 (2.7

7 Note O Eule Type Itegals 7 Poof: To evaluate the above itegal, we expess S x] i its seies fo with the help of Eq.(. ad Geealized Mittag-Leffle fuctio i tes of seies fo give by Eq.(.9 ad the itechagig the ode of itegatio ad suatio, we get: I /] ( k k k,,k u u (γ kl (γ kl z k z k Γ(λ + k ρ + + k ρ (k!(k! x s+λk+µk x s+λk+µk ( x x t+ηk+δk++δk dx dx (2.9 Futhe usig the foula give i equatio (.25, the above equatio (2.9 educed to the followig fo /] k k,,k ( k (γ kl (γ kl z k z k Γ(λ + k ρ + + k ρ (k!(k! Γ(s + λ k + µ k Γ(s + λ k + µ k Γ(t + ηk + δ k + + δ k ( Γ (s i + λ i k + µ i k i + δ i k i + t + ηk i (2. /] k Γ(s + λ kγ(s + λ kγ(t + ηk ( k (s + λ k µk (s + λ k µk (t + ηk ( δk ++δ k Γ (s i + λ i k + µ i k i + δ i k i + t + ηk /] k i k,,k (γ kl (γ kl z k z k Γ(λ + k ρ + + k ρ (k!(k! Γ(s + λ kγ(s + λ kγ(t + ηk ( k E (γj,sj+λjk,t+ηk,(lj,µj, δ j (ρ j,(µ j+δ j,λ,t+ηk+ (s z j+λ jk,, z ] (2. Thus, fo equatio (2. ad (2., we have ou equied esults (2.7 ad (2.8. The esult i Eq. (2. is established by takig the patial deivative o both sides of equatio Eq. (2.7 with espect to s. Equatio (2.2 ad (2.3 ae siilaly established by takig the patial deivative of Eq. (2.7 with espect to s ad t espectively. Eq. (2.4 is established by addig patial deivatives of the Eq. (2.7 with espect to s,, s ad t. To establish Eq. (2.5 fist addig the patial deivatives of Eq. (2.7 with espect to s,, s ad the subtact patial deivative of Eq. (2.7 with espect to t. To establish Eq. (2.6 fist takig patial deivative of Eq. (2.7 with espect to t ad the subtact patial deivatives of Eq. (2.7 with espect to s,, s. Lea 2.8 If a, c, ζ, ξ C +, b, d C +,, N5, Lea 2.], the + L (a,b (ξ; x L (c,d (ζ; x h h k Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( x h Γ (ξ ( k + Γ (ak + b + Γ (c (h k + d +. (2.2 Theoe 2.9 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j +

8 8 Meha Chad ad Eauel Guaiglia λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the L (a,b (ξ; xp u u 2 x xp ( x x q L (c,d ( x x t S xp ( x x q (ρ j,λ z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx h h k Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( h Γ (ξ ( k + Γ (ak + b + Γ (c (h k + d + /] k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. xλ ( x x η] (2.3 Poof: Fist we deote LHS of equatio (2.3 by I, the applyig the esult give i equatio (2.2, we have: I h h k Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( h Γ (ξ ( k + Γ (ak + b + Γ (c (h k + d + u u ( x x t+qh S x s+ph x s2+p2h 2 x s+ph xλ ( x x η] (ρ j,λ z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx, (2.4 ow applyig the esult give i equatio (2.8, we have the equied esult (2.3 afte little siplificatio. Theoe 2. If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the L (a,b (ξ; xp u u ( x x q L (c,d xp 2 x ( x x t S xp ( x x q xλ ( x x η] (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx Γ (a + b + Γ (c + d + h ( h Γ (ζ + Γ (ξ (σ h h ( ζ(h k ξk ( ζ ζ(h k ( ξ ξk + k Γ (ak + b + Γ (c (h k + d + /] k h k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. (2.5 Poof: Let ξ, ζ N ad usig (.25, the equatio (2.3 educed to the equied esult (2.5afte little siplificatio.

9 Note O Eule Type Itegals 9 3 Special Cases By applyig the ou esult give i equatio (2.8, (2.3 ] ad (2.5 to the case of Heite polyoials (Sivastava ad Sigh, 983 2] by settig S(x 2 x /2 H 2 i which 2; A x,k ( k, we have the above equied esult. Coollay 3. If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: H 2 xλ u u ( x x η 2 x ( x x t xλ ( x x η] /2 (ρ j,λ z (x µ (3. ( x x δ,, z (x µ ( x x δ ]dx dx /2] k Γ(s + λ kγ(s + λ kγ(t + λk ( 2k ( k E (γj,sj+λjk,t+ηk,(lj,µj, δ j (ρ j,(µ j+δ j,λ,t+ηk+ (s z j+λ jk,, z ]. Coollay 3.2 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the H 2 xλ L (c,d u u ( x x η 2 x L (a,b ( x x t xλ ( x x η] /2 xp ( x x q (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx /2] k h h k xp ( x x q Γ (a + b + Γ (c + d + ( h Γ (h k + Γ (ζ ( h + k + Γ (k + Γ (ξ ( k + Γ (ak + b + Γ (c (h k + d + Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( 2k ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. (3.2 Coollay 3.3 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j +

10 Meha Chad ad Eauel Guaiglia λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the H 2 xλ u u ( x x η 2 x L (a,b ( x x t xλ ( x x η] /2 xp ( x x q L (c,d xp ( x x q (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx Γ (a + b + Γ (c + d + h ( h Γ (ζ + Γ (ξ (σ h h ( ζ(h k ξk ( ζ ζ(h k ( ξ ξk + k Γ (ak + b + Γ (c (h k + d + /2] k h k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( 2k ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. By applyig the ou esult give i equatio (2.8, (2.3 ad (2.5 to ( the case of Lague polyoials (Sivastava ad Sigh, 983 2] by settig S(x L (α + α x] i which ;, we have the followig (α + k esults. (3.3 Coollay 3.4 If t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have: u u 2 x ( x x t L (α (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx ] k Γ(s + λ kγ(s + λ kγ(t + λk ( k ( + α xλ ( x x η] δj E (γj,sj+λjk,t+ηk,(lj,µj, (α + (ρ j,(µ j+δ j,λ,t+ηk+ (s z j+λ jk,, z ]. k Coollay 3.5 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the (3.4 L (a,b (ξ; xp u u 2 x xp ( x x q L (c,d ( x x t L (α xp ( x x q (ρ j,λ z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx h h k Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( h Γ (ξ ( k + Γ (ak + b + Γ (c (h k + d + ] k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. xλ ( x x η] ( + α (α + k (3.5 Coollay 3.6 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j +

11 Note O Eule Type Itegals λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the L (a,b (ξ; xp u u ( x x q L (c,d xp 2 x ( x x t L (α xp ( x x q xλ ( x x η] (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx Γ (a + b + Γ (c + d + h ( h Γ (ζ + Γ (ξ (σ h h ( ζ(h k ξk ( ζ ζ(h k ( ξ ξk + k Γ (ak + b + Γ (c (h k + d + ] k h k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. ( + α (α + k (3.6 If we take ρ ρ ad l l the Geealized Mittag-Leffle fuctio educed to the cofluet hypegeoetic seies 22, p.34, Eq. (.4(8], the the esults fo equatio (2.8, (2.3 ad (2.5 educed to the followig fo. Coollay 3.7 If t, η, λ, s j, λ j, γ j, δ j, µ j C; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > (j, 2,, ; λ / Z, the we have: u u 2 x ( x x t S xλ ( x x η] φ ( 2 γ,, γ ; λ; z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx /] k Γ(s + λ kγ(s + λ kγ(t + λk ( k E (γj,sj+λjk,t+ηk,(,µj, δ j (,(µ j+δ j,λ,t+ηk+ (s z j+λ jk,, z ]. (3.7 Coollay 3.8 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, δ j, µ j C; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; (j, 2,, ; λ / Z, the L (a,b (ξ; xp u u 2 x xp ( x x q L (c,d ( x x t S xp ( x x q xλ ( x x η] φ ( 2 γ,, γ ; λ; z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx h h k Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( h Γ (ξ ( k + Γ (ak + b + Γ (c (h k + d + /] k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(,µ j, δ j (,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. (3.8 Coollay 3.9 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, δ j, µ j C; R(γ j > ; R(s j + λ j k + µ j k j >

12 2 Meha Chad ad Eauel Guaiglia ; R(t + ηk + δ k + + δ k > ; (j, 2,, ; λ / Z, the L (a,b (ξ; xp φ ( u u ( x x q L (c,d xp 2 x ( x x t S xp ( x x q xλ ( x x η] 2 γ,, γ ; λ; z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx Γ (a + b + Γ (c + d + h ( h Γ (ζ + Γ (ξ (σ h h ( ζ(h k ξk ( ζ ζ(h k ( ξ ξk + k Γ (ak + b + Γ (c (h k + d + /] k h k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(,µ j, δ j (,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. (3.9 If we take the above esult i equatios (2.8, (2.3 ad (2.5 educed to the followig fo. Coollay 3. If t, η, λ, s, λ, γ, ρ, δ, µ C; R(ρ > ; R(γ > ; R(s + λ k + µ k > ; R(t + ηk + δ k > ; l N; λ / Z, the we have: ( x t S /] k ( x η] E (γ,(l (ρ,λ z (x µ ( x δ ]dx Γ(s + λ kγ(t + λk ( k E (γ,s+λk,t+ηk,(l,µ,δ (ρ z,(µ +δ,λ,t+ηk+(s +λ k ]. Coollay 3. If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s, λ, γ, ρ, δ, µ C; R(ρ > ; R(γ > ; R(s + λ k + µ k > ; R(t + ηk + δ k > ; l N; λ / Z, the ( x t S ( x η] L (a,b E (γ,(l (ρ z,λ (x µ ( x δ ]dx h h k ( x q L (c,d ( x q Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( h /] Γ (ξ ( k + Γ (ak + b + Γ (c (h k Γ(s + p h + λ kγ(t + qh + ηk ( k + d + E (γ,s+ph +λ k,t+ηk,(l,µ,δ (ρ,(µ +δ,λ,t+qh +ηk+(s +p h +λ k z ]. Coollay 3.2 If a, c, ζ, ξ C +, b, d C +,, N; t, η, λ, s, λ, γ, ρ, δ, µ C; R(ρ > ; R(γ > ; R(s + λ k + µ k > ; R(t + ηk + δ k > ; l N; λ / Z, the ( x t S ( x η] L (a,b E (γ,(l (ρ,λ z (x µ ( x δ ]dx Γ (a + b + Γ (c + d + Γ (ζ + Γ (ξ + /] k h (σ h h k k ( x q L (c,d ( x q ( h h ( ζ(h k ξk ( ζ ζ(h k ( ξ ξk k Γ (ak + b + Γ (c (h k + d + Γ(s + p h + λ kγ(t + qh + ηk ( k E (γ,sj+ph +λ k,t+ηk,(l,µ,δ (ρ,(µ +δ,λ,t+qh +ηk+ (s +p h +λ z k ]. O settig l j γ j ρ j λ ; the ultivaiable Mittag-Leffle fuctio educed to expoetial fuctio i.e. E,, (z E,(z exp(z ; ou ai esults established i equatios (2.8, (2.3 ad (2.5, educed to the followig (3. (3. (3.2

13 Note O Eule Type Itegals 3 fo: Coollay 3.3 If t, η, s, λ, δ, µ C; R(s + λ k + µ k > ; R(t + ηk + δ k >, the we have: ( x t S /] k ( x η] exp ( z (x µ ( x δ dx Γ(s + λ kγ(t + λk ( k E (,s+λk,t+ηk,(,µ,δ (,(µ z +δ,,t+ηk+(s +λ k ]. (3.3 Coollay 3.4 If a, c, ζ, ξ C +, b, d C +,, N; t, η, s, λ, δ, µ C; R(s + λ k + µ k > ; R(t + ηk + δ k >, the we have: ( x t S exp ( z (x µ ( x δ dx ( x η] L (a,b h h k ( x q L (c,d ( x q Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( h /] Γ (ξ ( k + Γ (ak + b + Γ (c (h k Γ(s + p h + λ kγ(t + qh + ηk ( k + d + E (,s+ph +λ k,t+ηk,(,µ,δ (,(µ +δ,,t+qh +ηk+(s +p h +λ k z ]. k (3.4 Coollay 3.5 If a, c, ζ, ξ C +, b, d C +,, N; t, η, s, λ, δ, µ C; R(s + λ k + µ k > ; R(t + ηk + δ k >, the we have: ( x t S Γ (a + b + Γ (c + d + Γ (ζ + Γ (ξ + /] k ( x η] L (a,b h (σ h h k ( x q L (c,d (ξ; xp ( x q exp ( z (x µ ( x δ dx ( h h ( ζ(h k ξk ( ζ ζ(h k ( ξ ξk k Γ (ak + b + Γ (c (h k + d + (3.5 Γ(s + p h + λ kγ(t + qh + ηk ( k E (,sj+ph +λ k,t+ηk,(,µ,δ (,(µ +δ,,t+qh +ηk+ (s +p h +λ z k ]. O settig ; λ γ j l j ; ρ 2; the ultivaiable Mittag-Leffle fuctio educed to the hypebolic fuctio i.e. E, 2, z2 ] E 2, z 2 ] cosh(z; ou ai esults established i equatios (2.8, (2.3 ad (2.5, educed to the followig fo: Coollay 3.6 If t, η, s, λ, δ, µ C; R(s + λ k + µ k > ; R(t + ηk + δ k >, the we have: ( x t S /] k ( x η] cosh ( z (x µ ( x δ dx Γ(s + λ kγ(t + λk ( k E (,s+λk,t+ηk,(,µ,δ (2,(µ z +δ,,t+ηk+(s +λ k ]. (3.6 Coollay 3.7 If a, c, ζ, ξ C +, b, d C +,, N; t, η, s, λ, δ, µ C; R(s + λ k + µ k > ; R(t + ηk +

14 4 Meha Chad ad Eauel Guaiglia δ k >, the we have: ( x t S ( x η] L (a,b cosh ( z (x µ ( x δ dx h h k ( x q L (c,d ( x q Γ (a + b + Γ (c + d + Γ (h k + Γ (ζ ( h + k + Γ (k + ( h /] Γ (ξ ( k + Γ (ak + b + Γ (c (h k Γ(s + p h + λ kγ(t + qh + ηk ( k + d + E (,s+ph +λ k,t+ηk,(,µ,δ (2,(µ +δ,,t+qh +ηk+(s +p h +λ k z ]. k (3.7 Coollay 3.8 If a, c, ζ, ξ C +, b, d C +,, N; t, η, s, λ, δ, µ C; R(s + λ k + µ k > ; R(t + ηk + δ k >, the we have: ( x t S Γ (a + b + Γ (c + d + Γ (ζ + Γ (ξ + /] k ( x η] L (a,b h (σ h h k ( x q L (c,d (ξ; xp ( x q cosh ( z (x µ ( x δ dx ( h h ( ζ(h k ξk ( ζ ζ(h k ( ξ ξk k Γ (ak + b + Γ (c (h k + d + (3.8 Γ(s + p h + λ kγ(t + qh + ηk ( k E (,sj+ph +λ k,t+ηk,(,µ,δ (2,(µ +δ,,t+qh +ηk+ (s +p h +λ z k ]. O settig ζ ξ, the esult i Eq. (2.5 educed to the followig fo: Coollay 3.9 If a, c C +, b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the L (a,b (; xp u u ( x x q L (c,d xp 2 x ( x x t S (; xp xp ( x x q xλ ( x x η] (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx Γ (a + b + Γ (c + d + h ( h Γ ( + Γ ( (σ h h ( (h k k ( (h k ( k + k Γ (ak + b + Γ (c (h k + d + /] k h k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. (3.9 O settig a c ξ ζ ad usig Eq.(.4, we have L,b (; x Z (,b (x; ;the esult i Eq. (2.5 educed to the followig fo: Coollay 3.2 If b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the we have:

15 Note O Eule Type Itegals 5 Z (,b (xp xp u u ( x x q ; Z (,d 2 x ( x x t S (xp xp ( x x q ; xλ ( x x η] (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx Γ (a + b + Γ (c + d + h ( h Γ ( + Γ ( (σ h h ( (h k k ( (h k ( k + k Γ (ak + b + Γ (c (h k + d + /] k h k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. (3.2 O settig a c ; ξ ζ ;the esult i Eq. (2.5 educed to the followig fo: Coollay 3.2 If b, d C +,, N; t, η, λ, s j, λ j, γ j, ρ j, δ j, µ j C; R(ρ j > ; R(γ j > ; R(s j + λ j k + µ j k j > ; R(t + ηk + δ k + + δ k > ; l j N(j, 2,, ; λ / Z, the u u 2 x σ(x p xp ( x x q ] ( x x t S (ρ j,λ z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx h (σ h ( h /] k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. xλ ( x x η] Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k (3.2 Poof: Put a c ; ξ ζ ; the esult i Eq. (2.5, we have: L (,b (; xp u u ( x x q L (,d xp 2 x ( x x t S (; xp xp ( x x q (ρ z j,λ (x µ ( x x δ,, z (x µ ( x x δ ]dx dx ( ] h Γ ( + Γ ( + /] k h (σ h h k k ( (h k ( k Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. xλ ( x x η] (3.22 Applyig the esult (.25 i RHS of Eq. (3.22, we get:

16 6 Meha Chad ad Eauel Guaiglia L (,b (; xp u u ( x x q L (,d xp 2 x ( x x t S (; xp xp ( x x q (ρ j,λ z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx h ( /] (σ h h Γ( + Γ( + k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. xλ ( x x η] Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k (3.23 Also applyig the case (. ad usig Eq. (.25-(.28, we get: L (,β (x Γ( + h ( h k k ( x k Γ( + ( x (3.24 Fo Eq. (3.23 ad (3.24, we have: u u 2 x σ(x p xp ( x x q ] Γ( + Γ( + ( x x t S (ρ j,λ z (x µ ( x x δ,, z (x µ ( x x δ ]dx dx h ( /] (σ h h Γ( + Γ( + k E (γj,sj+pjh +λ jk,t+ηk,(l j,µ j, δ j (ρ j,(µ j+δ j,λ,t+qh +ηk+ (s j+p jh +λ jk z,, z ]. xλ ( x x η] Γ(s + p h + λ kγ(s + p h + λ kγ(t + qh + ηk ( k (3.25 Replacig + by, we get esult (3.2. Refeeces ] Agawal, R.P., (953. A popos due ote de M. Piee Hubet, C.R. Acad. Sci., Pais, 236, ] Dzhebashya, M.M., (966. Itegal Tasfos ad Repesetatios of Fuctios i the Coplex Plae, Nauka, Moscow, (i Russia. 3] Edelyi, A., Magus,W., Obehettige, F. ad Ticoi, F. G. (955. Highe Tascedetal Fuctios, McGaw - Hill, Vol. 3, New Yok, Tooto ad Lodo. 4] Gauta, S.(28. Ivestigatios i Factioal Diffeetial Opeatos of Abitay Ode ad thei Applicatios to Special Fuctios of Oe ad Seveal Vaiables, Ph.D. Thesis, Uivesity of Kota, Kota, Idia. 5] Hubet, P., (953. Quelques esultats etifs a la foctio de Mittag-Leffle, C.R. Acad. Sci. Pais, 236, ] Hubet, P. ad Agawal, R.P., (953. Su la foctio de Mittag-Leffle et quelques ues de ses geealizatios, Bull. Sci. Math., (Se.II, 77, ] Hilfe, R. (ed., (2. Applicatios of Factioal Calculus i Physics, Wold Scietific, Sigapoe. 8] Lag, K.R., (999a. Astophysical Foulae, Vol. : Radiatio, Gas Pocesses ad High-eegy Astophysics, 3d editio, evised editio, Spige-Velag, New Yok.

17 Note O Eule Type Itegals 7 9] Lag, K.R., (999b. Astophysical Foulae, Vol. 2: Space, Tie, Matte ad Cosology, Spige-Velag, New Yok. ] Mittag-Leffle, G.M., (93. Ue geealisatio de litegale de Laplace-Abel, C.R. Acad. Sci. Pais, (Se. II, 37, ] Mittag-Leffle, G.M., (95. Su la epesetatio aalytiqie due foctio oogee (ciquiee ote, Acta Matheatica, 29, -8. 2] Pabhaka, T.R. (97. A Sigula itegal equatio with a geealized Mittag-Leffle fuctio i the keel. Yokohaa Math. J., Vol. 9, pp ] Pabhaka, T.R. ad Sua, R. (978. Soe esults o the polyoials L α,β (x, Rocky Moutai J. Math. 8, o. 4, ] Raiville, E.D. (96. Special Fuctios, Macilla, New Yok. 5] Pavee Agawal, Shipli Jai ad Meha Chad,(25. Cetai Itegals Ivolvig Geealized Mittage-Leffle Fuctio, Poceedigs of the Natioal Acadey of Scieces, Idia - Sectio A /25(Accepted. 6] Saxea, R.K., Kalla, S.L. ad Saxea, R.,(2. Multivaiable aalogue of geealized Mittag-Leffle fuctio, Itegal Tas. Special Fu., Vol. 22,No. 7, ] Saxea, R.K., Mathai, A.M. ad Haubold, H.J., (22. O factioal kietic equatios, Astophysics ad Space Sciece, 282, ] Spaie, J. ad Oldha, K.B., (987. A Altas of fuctios, Heisphee, Washigto DC, Spige, Beli. 9] Sivastava, H.M., (972. A cotou itegal ivolvig Foxs H-fuctio, Idia J. Math., 4: -6. 2] Sivastava, H.M. ad N.P. Sigh, (983. The itegatio of cetai poducts of the Multivaiable H-fuctio with a geeal class of polyoials, Red. Cic. Mat. Paleo 2(32: ] Sivastava, H.M., (985. A ultiliea geeatig fuctio fo the Kohause sets of bi-othogoal polyoials suggested by the Laguee polyoials, Pacific J. Math. 7(: ] Sivastava, H.M. ad Kalsso, P.W. (985. Multiple Gaussia hypegeoetic Seies. Ellis Howood Seies: Matheatics ad its Applicatios. Ellis Howood Ltd., Chicheste; Halsted Pess Joh Wiley & Sos, Ic.], New Yok. 23] Sivastava, H.M. ad Toovski, Z.(29. Factioal calculus with a itegal opeato cotaiig a geealized Mittag-Leffle fuctio i the keel. Appl. Math. Coput. 2(, ] Shukla, A.K., Pajapati, J.C. ad Salehbhai, I.A. (29. O a set of polyoials suggested by the faily of Kohause polyoial, It. J. Math. Aal. 3, o. 3-6, ] Shukla, A.K. ad Pajapati, J.C. (27. O a geealizatio of Mittag-Leffle fuctio ad its popeties, J. Math. Aal. Appl., 336(2, ] Wia, A., (95. Übe de Fudaetal satz i de Theoie de Fucktioe, E α(x, Acta Matheatica, 29, 9-2.