Marichev-Saigo-Maeda Fractional Integral Operators Involving the Product of Generalized Bessel-Maitland Functions

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1 Bol. Soc. Para. Mat. 3s. v :????. c SPM ISSN o lie ISSN i press SPM: doi: /bspm Marichev-Saigo-Maeda Fractioal Itegral Operators Ivolvig the Product of Geeralized Bessel-Maitlad Fuctios D.L. Suthar, P. Agarwal ad Hafte Amsalu abstract: The aim of this paper is to evaluate two theorems for fractioal itegratio ivolvig Appell s fuctio F 3. due to Marichev-Saigo-Maeda, to the product of the geeralized Bessel-Maitlad fuctio. The results are expressed i terms of the multivariable geeralized Lauricella fuctios. Correspodig assertios i terms of Saigo, Erdélyi-Kober, Riema-Liouville, ad Weyl type of fractioal itegrals are also preseted. Some iterestig special cases of our two mai results are preseted. Further, we poit out also their relevace. Key Words:Geeralized Fractioal itegrals, Geeralized Bessel-Maitlad fuctio, geeralized Lauricella series i several variables, Appell fuctio-f 3.. Cotets 1 Itroductio, Defiitio ad prelimiaries 1 2 Left-Side Fractioal Itegratio of Geeralized Bessel-Maitlad Fuctios 4 3 Right-Side Fractioal Itegratio of Geeralized Bessel-Maitlad Fuctios 6 4 Cosequece Results ad Cocludig Remarks 9 1. Itroductio, Defiitio ad prelimiaries The fractioal calculus is ow a day s oe of the most fast growigsubject of mathematical aalysis. It is a field of applied mathematics that deals with derivatives ad itegrals of arbitrary orders. The fractioal itegral operator ivolvig various special fuctios has foud cosiderable importace ad applicatios i various sub-fields of applicable mathematical aalysis. May applicatios of fractioal calculus ca be foud i turbulece ad fluid dyamics, stochastic dyamical system, plasma physics ad oliear cotrol theory, image processig, oliear biological systems, astrophysics, ad i quatum mechaics. Sice last four decades, a umber of workers like Agarwal 1-3, Agarwal ad Jai 5, Baleau 8, Baleau ad Mustafa 9, Baleau et al , Kalla 14, Kalla ad Saxea 15, Kilbas 16, Kilbas ad Sebastia 17, Kiryakova 18-19, Love 20, 2010 Mathematics Subject Classificatio: 33C10. Submitted May 18, Published November 18, Typeset by B S P M style. c Soc. Para. de Mat.

2 2 D.L. Suthar, P. Agarwal ad Hafte Amsalu McBride 22, Purohit ad Kalla 25 ad Saigo 29-30, so forth have studied, i depth, the properties, applicatios, ad differet extesios of various operators of fractioal calculus. A detailed accout of geeralized fractioal calculus operators alog with their properties ad applicatios ca be foud i the research moographs by Kiryakova 18, Miller ad Ross 23, ad so forth. The computatio of fractioal derivatives ad fractioal itegrals of special fuctios of oe ad more variables is importat from the poit of view of the usefuless of these results i the evaluatio of geeralized itegrals ad the solutio of differetial ad itegral equatios. Motivated by these aveues of applicatios, a remarkably large umber of fractioal itegral formulas ivolvig a variety of special fuctios have bee developed by may authors see, e.g., 7, 26-28, 34, 35. Fractioal itegratio formulae for the Bessel fuctio ad geeralized Bessel fuctios are give recetly by Kilbas ad Sebastia 17, Malik et al. 21, Purohit et al. 28 ad Saxea et al. 32. Here, we aim at presetig compositio formula of Marichev-Saigo-Maeda fractioal itegral operators ad the product of geeralized Bessel-Maitlad fuctio, which are expressed i terms of the multivariable geeralized Lauricella fuctios. Some iterestig special cases of our mai results are also cosidered. O accout of success of the Saigo operators 29-30, i their study o various fuctio spaces ad their applicatio i the itegral equatio ad differetial equatios, Saigo ad Maeda 31 itroduced the followig geeralized fractioal itegral ad differetial operators of ay complex order with Appell fuctio F 3. i the kerel, as follows: Let α,α,β,β,γ C ad x > 0, the the geeralized fractioal calculus operators the Marichev-Saigo-Maeda operators ivolvig the Appell fuctio, or Hor s F 3 -fuctio are defied by the followig equatios: where x 0 x 0+ f x = x α Γγ x t γ 1 t α F 3 α,α,β,β ;γ;1 t x,1 x ftdt, Reγ > 0, 1.1 t f x = x α Γγ t x γ 1 t α F 3 α,α,β,β ;γ;1 x t,1 t ftdt, Reγ > 0, 1.2 x F 3 α,α,β,β ;γ,x;y = m,=0 αm α β m β x m y, max{ x, y < 1}, γ +m m!! is kow as Appell fuctio kow also as Hor fuctio.

3 Marichev-Saigo-Maeda Fractioal Itegral Operators... 3 Followig Saigo ad Maeda 31, the image formulas for a power fuctio, uder operators 1.1 ad 1.2, are give by: 0+ x ρ 1 ρ, ρ+γ α α β, ρ+β α x = Γ ρ+β, ρ+γ α α, ρ+γ α x ρ α α +γ 1, β 1.3 where Rρ > max { 0, Rα+α +β γ, Rα β } ad Rγ > 0. x ρ 1 1 ρ β, 1 ρ+α+β γ, 1 ρ γ +α+α x = Γ 1 ρ, 1 ρ+α β, 1 ρ+α+α +β γ, x ρ+γ α α 1, 1.4 where Rρ < 1+mi { R β, Reα+α γ, Rα+β γ } ad Reγ > 0. represetig the fractio of may Gamma fuc-... Here we used the symbol Γ... tios. We ivestigate compositio of itegral trasforms 1.1 ad 1.2 with the product of geeralized Bessel-Maitlad fuctios J v,q µ,γ z, which is defied by Pathak 24 as follow: J v,q µ,γ z = γ qm z m Γv +µm+1 m!, 1.5 S p,p 1,p 2,...,p q,q 1,q 2,...,q m=0 where µ,v,γ C, Rµ 0, Rv 1, Rγ 0 ad q 0, 1 N, γ qm is kow as geeralized Pochhammer symbol, is defied as γ qm = Γγ+qm Γγ. Also, we show that the compositio is expressed i terms of the multivariable geeralized Lauricella fuctios due to Srivastava ad Daoust 33 is a geeralizatio of the Wright fuctio p ψ q i several variables 13 ad defied by a p : αp 1,..., α c 1 p : : γ 1 p1 p 1 ;...;c p1 : γ p 1 ; b q : β 1 q,..., β p : = m 1,m 2,...,m =0 d 1 : δ 1 q1 q1 ;...;d q1 : z 1,...,z δ q1 ; Am 1,m 2,...,m z mj j m j!, 1.6 where p k=1 a Γ j + m kα k pk j k=1 Am 1,m 2,...,m = q k=1 b Γ j + Γ c j + m kγ k j m kβ k qk j k=1 k=1 Γ d j +, m kδ k j 1.7 Thecoefficietα k j j = 1,2,...,p,βk j j = 1,2,...,q,γ k j j = 1,2,...,p kadδ k j j = 1,2,...,q k for all k = 1,...,, are real ad positive ad a p meas the arrayof p-parameters a 1,...,a p ; with similar iterpretatios for b p, γ 1 p 1, α 1 p1 ad so forth.

4 4 D.L. Suthar, P. Agarwal ad Hafte Amsalu The paper is orgaized as follow. The compositio formula of Marichev-Saigo- Maeda fractioal itegral operators 1.1 ad 1.2 with the product of geeralized Bessel-Maitlad fuctio 1.5 is proved i terms of the multivariable geeralized Lauricella fuctios 1.6 i Sectios 2 ad 3, respectively. The correspodig results for the correspodig assertios i terms of Saigo, Erdélyi-Kober, Riema- Liouville ad Weyl type of fractioal itegrals are also preseted i Sectios 2 ad 3. Special cases givig compositios of fractioal itegrals with the product of Bessel fuctios ad cocludig remarks are cosidered i Sectio Left-Side Fractioal Itegratio of Geeralized Bessel-Maitlad Fuctios I this sectio, we establish image formulas for the product of geeralized Bessel- Maitlad fuctio ivolvig left sided of Marichev-Saigo-Meada fractioal itegral operator, i term of the multivariable geeralized Lauricella fuctios. These formulas are give by the followig theorem: Theorem 2.1. Let α,α,β,β,σ,λ,γ,ν j, q j, µ j, k j, b C ad satisfyig the iequalities, Rγ > 0, Rσ > 0, Rσ > max { 0,Rα+α +β γ,r α β } ad R ν j +1 > 0 the for x > 0, ax b < 1 oe has 0+ t σ 1 b at λ a j t ρ 1 j x = xσ+γ α α Γλ b λ 1 A, Γk j S3;1;...;1;1 0 B, C, D, E : F : G; H; a 1 x ρ 1,..., a x ρ ax,, 2.1 b where A, B, C, D, E, F, G ad H are give by the followig A = σ : ρ 1,ρ 2,...,ρ,1, 2.2 B = σ +β : ρ 1,ρ 2,...,ρ,1, 2.3 C = σ +γ α α β : ρ 1,ρ 2,...,ρ,1, 2.4 D = σ +γ α α : ρ 1,ρ 2,...,ρ,1, 2.5 E = σ +β α : ρ 1,ρ 2,...,ρ,1, 2.6 F = σ +γ α β : ρ 1,ρ 2,...,ρ,1, 2.7 G = k 1 : q 1 ;...;k : q ; λ : 1, 2.8 H = v 1 +1 : µ 1 ;...;v +1 : µ ;

5 Marichev-Saigo-Maeda Fractioal Itegral Operators... 5 Proof: For the sake of coveiece, let the left-had side of the 2.1 be deoted by F. Usig defiitio 1.5 ad biomial expasio, amely, we fid F = b at λ = b λ 0+ = 0+ m=0 λ m m! m at, b t σ 1 b at λ m j=0 at b < a j t ρ j x t m σ 1 b λ λ m at m! b m=0 k j qjm j a j t ρ m j j x, 2.11 m j!γv j +µ j m j +1 Followig the covergece coditio of Theorem 2.1, for ay k N 0, R σ +m+ ρ jm j R σ + ρ jm j > max { 0,Rα+α +β γ,r α β }, 2.12 ad chagig the order of itegratio ad summatio, we obtai F = b λ m,m 1,m 2,...,m =0 k 1 q1m 1 a 1 m1 λ m a Γv 1 +µ 1 m 1 +1 m!m 1! b... k qm a m 0+ t σ+m+ρ 1 m1+...+ρ m 1 x, 2.13 Γv +µ m +1 m! Now o applyig 1.3 with ρ replaced by σ+m+ρ 1 m ρ m, we obtai F = xσ+γ α α 1 Γλ b λ Γ Γ 1 Γk j m,m 1,m 2,...,m =0 σ +m+γ α α β + ρ jm j Γ σ +m+γ α α ρ jm j Γ m Γ σ +m+ ρ jm j Γ σ +m+β + ρ jm j σ +m α +β + ρ jm j σ +m+γ α β + ρ jm j Γk 1 +q 1 m 1 Γv 1 +µ 1 m Γk +q m Γv +µ m +1 Γλ+m

6 6 D.L. Suthar, P. Agarwal ad Hafte Amsalu a 1x ρ 1 m 1 m 1!... a x ρ m m! ax b m! m I accordace with 1.6, gives the required result 2.1. This completed the proof of the Theorem 2.1. Now, we preset some special cases of Theorem 2.1 are give below: O settig α = 0 i Theorem 2.1, we get the followig Saigo fractioal itegral image of the product of geeralized Bessel-Maitlad fuctio. Corollary 2.2. Let α,β,β,σ,λ,γ,ν j, q j, µ j, k j, b C ad x > 0, ax b < 1 satisfyig the iequalities, Rγ > 0, Rσ > 0, R ν j +1 > 0 the there holds the results: I γ,α γ, β 0+ t σ 1 b at λ a j t ρ j x = xσ+γ α 1 Γλ b λ 1 A, Γk j S2;1;...;1;1 0 D, C : F : G; H; a 1 x ρ 1,..., a x ρ ax,. b 2.15 where A, C, D, F, G ad H are give by 2.2, 2.4, 2.5, 2.7, 2.8 ad 2.9 respectively. Agai, o lettig α = 0 ad α = 0 i Theorem 2.1, we get the Riema-Liouville fractioal image of the product of geeralized Bessel-Maitlad fuctio; asserted by the followig corollary. Corollary 2.3. Let α,β,β,σ,λ,γ,ν j, q j, µ j, k j, b C ad x > 0, ax b < 1 satisfyig the iequalities, Rγ > 0, Rσ > 0, R ν j +1 > 0 the there holds the results: I γ 0+ t σ 1 b at λ a j t ρ j x = xσ+γ 1 Γλ b λ 1 Γk j S1;1;...;1; 1 A : 0 D : G; H; a 1 x ρ 1,..., a x ρ ax, b where A, D, G ad H are give by 2.2, 2.5, 2.8 ad 2.9 respectively.

7 Marichev-Saigo-Maeda Fractioal Itegral Operators Right-Side Fractioal Itegratio of Geeralized Bessel-Maitlad Fuctios I this sectio, we establish image formulas for the product of geeralized Bessel- Maitlad fuctio ivolvig right sided of Marichev-Saigo-Meada fractioal itegral operator, i term of the multivariable geeralized Lauricella fuctios. These formulas are give by the followig theorem: Theorem 3.1. Let α,α,β,β,σ,λ, γ, ν j, q j, µ j, k j, b C ad satisfyig the iequalities, Rγ > 0, Rσ < 1 + mi { R β,rα+α γ,r α+β γ } Rσ > 0, ad R ν j +1 > 0 the for x > 0, a < 1 oe has t σ 1 b a/t λ bx a j /t ρ 1 j x = xσ+γ α α Γλ b λ 1 A0, Γk j S3;1;...;1;1 3;1;...;1;0 B 0, C 0, D 0, E 0 : F 0 : G 0 ; H 0 ; a 1 x ρ 1,..., a x ρ, a, 3.1 bx where A 0, B 0, C 0, D 0, E 0 ad F 0 are give by the followig: A 0 = 1 σ β : ρ 1,ρ 2,...,ρ, 1, 3.2 B 0 = 1 σ : ρ 1,ρ 2,...,ρ, 1, 3.3 C 0 = 1 σ γ +α+α : ρ 1,ρ 2,...,ρ, 1, 3.4 D 0 = 1 σ +α β : ρ 1,ρ 2,...,ρ, 1, 3.5 E 0 = 1 σ +α+β γ : ρ 1,ρ 2,...,ρ, 1, 3.6 F 0 = 1 σ +α+α +β γ : ρ 1,ρ 2,...,ρ, 1, 3.7 ad G, H are give by 2.8 ad 2.9, respectively. Proof: For coveiece, Let the left-had side of the 3.1 be deoted by F. Usig defiitio 1.5 ad biomial expasio, we fid F = t σ 1 b a/t λ a j /t ρ j x = t σ 1 b λ λ m a m m! bt m=0 k j qjm j a j /t ρ j m j x, 3.8 m j!γv j +µ j m j +1 m j=0

8 8 D.L. Suthar, P. Agarwal ad Hafte Amsalu Followig the covergece coditio of Theorem 3.1, for ay k N 0, R σ m ρ j m j R σ ρ j m j < 1+mi { R β,rα+α γ,r α+β γ }, 3.9 ad chagig the order of itegratio ad summatio, we obtai F = b λ m,m 1,m 2,...,m =0 k 1 q1m 1 a 1 m1 λ m a Γv 1 +µ 1 m 1 +1 m!m 1! b... k qm a m t σ m ρ 1 m1... ρ m 1 x, 3.10 Γv +µ m +1 m! Now o applyig 1.3 with ρ replaced by σ+m+ρ 1 m ρ m, we obtai F = xσ+γ α α 1 Γλ b λ 1 Γk j m,m 1,m 2,...,m =0 m Γ 1 σ β +m+ ρ jm j Γ 1 σ +m+ ρ j m j Γ 1 σ+α+β γ +m+ ρ jm j Γ 1 σ +α+α +β γ +m+ ρ j m j Γ 1 σ γ +α+α +m+ ρ jm j Γ 1 σ +α β +m+ ρ j m j Γk 1 +q 1 m 1 Γv 1 +µ 1 m Γk +q m Γv +µ m +1 Γλ+m a 1/x ρ 1 m 1 m 1!... a /x ρ m m! a m bx m! I accordace with 1.6, gives the required result 3.1. This completed the proof of the Theorem 3.1. Now, we preset some special cases of Theorem 3.1 are give below: O settig α = 0 i Theorem 3.1, we get the followig Saigo fractioal itegral image of the product of geeralized Bessel-Maitlad fuctio.

9 Marichev-Saigo-Maeda Fractioal Itegral Operators... 9 Corollary 3.2. Let α,β,β,σ,λ,γ,ν j, q j, µ j, k j, b C ad x > 0, a bx < 1 satisfyig the iequalities, Rγ > 0, Rσ > 0, R ν j +1 > 0 the there holds the results: = xσ+γ α 1 Γλ b λ I γ,α γ, β t σ 1 b a/t λ 1 A0, Γk j S2;1;...;1;1 2;1;...;1;0 B 0, C 0 : D 0 : a j /t ρ j x G; H; a 1 x ρ 1,..., a x ρ, a bx where A 0, B 0, C 0 ad D 0 are give by 3.2, 3.3, 3.4 ad 3.5 respectively. Agai, o lettig α = 0 ad α = 0 i Theorem 3.1, we get the Riema-Liouville fractioal image of the product of geeralized Bessel-Maitlad fuctio; asserted by the followig corollary. Corollary 3.3. Let α,β,β,σ,λ,γ,ν j, q j, µ j, k j, b C ad x > 0, a bx < 1 satisfyig the iequalities, Rγ > 0, Rσ > 0, R ν j +1 > 0 the there holds the results: = xσ+γ 1 Γλ b λ I γ t σ 1 b a/t λ 1 C0 : Γk j S1;1;...;1;1 1;1;...;1;0 B 0 : a j /t ρ j x G; H; a 1 x ρ 1,..., a x ρ, a bx where B 0, C 0, G ad H are give by 3.3, 3.4, 2.8 ad 2.9 respectively. 4. Cosequece Results ad Cocludig Remarks I this sectio, we briefly cosider aother variatio of the results derived i the precedig sectios. Bessel-Maitlad fuctios are importat special fuctios that appear widely i sciece ad egieerig. Bessel-Maitlad fuctios are oscillatory ad may be regarded as geeralizatios of Bessel fuctios. Further, it ca be easily see that for q = 1, γ = 1 ad v is replaced by v +σ ad z is replaced by z 2 /4, the geeralized Bessel-Maitlad fuctio 1.5 reduces to J v,σ µ z which is defied by Agarwal et al. 6, ad whe q = 0 the fuctio reduces to geeralized Bessel fuctio J v µ z defied by Agarwal 4. Similarly whe q = 0, µ = 1 ad z is replaced by z 2 /4, the geeralizedbessel-maitlad fuctio 1.5 reduces to Bessel s fuctio J v z of the first kid of order v, ad whe q = 0 ad v is replaced by v 1 ad z is replaced by z reduces to Wright fuctio φµ,v;z which is defied by defied by Choi et al. 12. Therefore, the results preseted i this paper are easily coverted i terms of the various special Bessel fuctios after some suitable parametric replacemet.

10 10 D.L. Suthar, P. Agarwal ad Hafte Amsalu The geeralized Bessel-Maitlad fuctio defied by 1.5, possess the advatage that a umber of Bessel fuctios, Mittag-Leffler fuctio, trigoometric fuctios ad hyperbolic fuctios happe to be the particular cases of this fuctio. Therefore, we coclude this paper with the remark that, the results deduced above are sigificat ad ca lead to yield umerous other fractioal itegrals ivolvig various Bessel fuctios ad trigoometric fuctios by the suitable specializatios of arbitrary parameters i the theorems. More importatly, they are expected to fid some applicatios to the solutios of fractioal differetial ad itegral equatios. The results thus derived i this paper are geeral i character ad likely to fid certai applicatios i the theory of special fuctios. Refereces 1. Agarwal, P., Further results o fractioal calculus of Saigo operators, Appl. Appl. Math., 7, 2, , Agarwal, P., Geeralized fractioal itegratio of the H fuctio, Matematiche Cataia, 67, 2, , Agarwal, P., Fractioal itegratio of the product of two multivariables H-fuctio ad a geeral class of polyomials, Advaces i Applied Mathematics ad Approximatio Theory, Spriger Proc. Math. Stat., Spriger, New York, 41, , Agarwal, P., Pathway fractioal itegral formulas ivolvig Bessel fuctio of the first kid, Adv. Stud. Cotemp. Math., 25, 1, , Agarwal, P., Jai, S., Further results o fractioal calculus of Srivastava polyomials, Bull. Math. Aal. Appl., 3, 2, , Agarwal, P., Jai, S., Chad, M., Dwivedi, S.K., Kumar, S., Bessel fuctios associated with Saigo-Maeda fractioal derivatives operators, J. Fract. Calc. Appl., 5, 2, , Agarwal, P., Purohit, S.D., The uified pathway fractioal itegral formulae, J. Fract. Calc. Appl., 4, 1, , Baleau, D., About fractioal quatizatio ad fractioal variatioal priciples, Commu Noliear Sci Numer Simul., 14, 6, , Baleau, D., Mustafa, O.G., O the global existece of solutios to a class of fractioal differetial equatios, Comput. Math. Appl., 59, 5, , Baleau, D., Mustafa, O.G., Agarwal, R.P., O the solutio set for a class of sequetial fractioal differetial equatios, J. Phys. A, 43, 38, Article ID , Baleau, D., Mustafa, O.G., O Rega, D., A uiqueess criterio for fractioal differetial equatios with Caputo derivative, Noliear Dyam., 71, 4, , Choi, J., Agarwal, P., Mathur, S., Purohit, S.D., Certai ew itegral formulas ivolvig the geeralized Bessel fuctios, Bull. Korea Math. Soc., 51, 4, , Exto, H., Multiple Hypergeometric Fuctios ad Applicatios, Foreword by L. J. Slater. Mathematics & its Applicatios. Ellis Horwood Ltd., Chichester; Halsted Press Joh Wiley & Sos, Ic., New York-Lodo-Sydey, 312, Kalla, S.L., Itegral operators ivolvig Fox s H-fuctio, Acta Mexicaa Ci. Tec., 3, , Kalla, S.L., Saxea, R.K., Itegral operators ivolvig hypergeometric fuctios, Math. Z., 108, , Kilbas, A.A., Fractioal calculus of the geeralized Wright fuctio, Fract. Calc. Appl. Aal., 8, 2, , 2005.

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12 12 D.L. Suthar, P. Agarwal ad Hafte Amsalu D.L. Suthar, Departmet of Mathematics, Wollo Uiversity, P.O.Box: 1145, Dessie, South Wollo, Amhara Regio, ETHIOPIA. address: ad P. Agarwal, Departmet of Mathematics, Aad Iteratioal College of Egieerig, Jaipur, , Rajastha, INDIA. address: ad Hafte Amsalu, Departmet of Mathematics, Wollo Uiversity, P.O. Box: 1145, Dessie, South Wollo, Amhara Regio, ETHIOPIA. address: