Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function

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1 Mishra et al. Cogent Mathematics : PURE MATHEMATICS RESEARCH ARTICLE Marichev-Saigo-Maeda ractional calculus operators Srivastava polynomials and generalized Mittag-Leler unction Received: 24 October 2016 Accepted: 29 March 2017 First Published: 21 April 2017 *Corresponding author: Vishnu Narayan Mishra Department o Applied Mathematics & Humanities Sardar Vallabhbhai National Institute o Technology Surat Gujarat India vishnunarayanmishra@gmail.com Reviewing editor: Hari M. Srivastava University o Victoria Canada Additional inormation is available at the end o the article Vishnu Narayan Mishra 1 * D.L. Suthar 2 and S.D. Purohit 3 Abstract: The aim o this paper is to evaluate our theorems or generalized ractional integral and derivative operators applied on the product o Srivastava polynomials and generalized Mittag-Leler unction. The results are expressed in terms o generalized Wright unction. Further we also point out their relevance with the known results. Subjects: Science; Mathematics & Statistics unction; Technology; Engineering & Technology Keywords: Marichev-Saigo-Maeda ractional operators; generalized Mittag-Leler unction; Srivastava polynomials; generalized Wright unction; Orthogonal polynomials and special unctions AMS subject classiications: 26A33; 33B15; 33C05; 33C99; 44A10 Vishnu Narayan Mishra ABOUT THE AUTHORS Vishnu Narayan Mishra received the PhD in Mathematics rom Indian Institute o Technology Roorkee. His research interests are in the areas o pure and applied mathematics. He has published more than 120 research articles in reputed international journals o mathematical and engineering sciences. He is a reeree and an editor o several international journals in rame o Mathematics. He guided many postgraduate and PhD students. Citations o his research contributions can be ound in many books and monographs PhD thesis and scientiic journal articles. D.L. Suthar is an Associate Proessor in the Department o Mathematics at Wollo University Dessie Amhara Region Ethiopia. His research interests include Special unctions Fractional calculus Integral transorms Basic Hypergeometric series and Mathematical physics. S.D. Purohit is Associate Proessor o Mathematics in Department o HEAS Mathematics at Rajasthan Technical University Kota Rajasthan India. His research interests include Special unctions Fractional Calculus Integral transorms Basic Hypergeometric Series Geometric Function Theory and Mathematical Physics. He has published more than 100 research papers in international esteemed journals. PUBLIC INTEREST STATEMENT The Mittag-Leler unctions are very useul almost in all areas o applied Mathematics that provides solutions to a number o problems ormulated in terms o ractional order dierential integral and dierence equations; thereore it has recently become a subject o interest or many authors in the ield o ractional calculus and its applications. In this paper we have evaluated our theorems or generalized ractional integral and derivative operators applied on the product o Srivastava polynomials and generalized Mittag-Leler unction and also point out their relevance with the known results The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. Page 1 o 11

2 Mishra et al. Cogent Mathematics : Introduction The Mittag-Leler unctions are important special unctions that provides solutions to number o problems ormulated in terms o ractional order dierential integral and dierence equations; thereore it has recently become a subject o interest or many authors in the ield o ractional calculus and its applications. For detailed account o ractional calculus operators along with their properties and applications one may reer to the research monographs by Kilbas Srivastava and Trujillo 2006 Kiryakova 1994 Miller and Ross 1993 Srivastava and Saigo 1987 Srivastava and Saxena 2001 and recent papers Mishra and Agarwal 2016 Mishra Agarwal and Sen 2016 Mishra and Sen 2016 Mishra Srivastava and Sen 2016 Purohit 2013 and Purohit and Kalla The Swedish mathematician Mittag-Leler 1903 introduced the unction E α z deined by: E α z n0 1 Γαn + 1 zn α C; Rα > 0 1 A urther two-index generalization o this unction was studied by Wiman 1905 as: E α β z n0 1 Γαn + β zn α β C 2 where Rα > 0 and Rβ > 0. Prabhakar 1971 introduced the generalization o Mittag-Leler unction E δ β γ z in the orm E δ β γ z n0 δ n Γβn + γn! zn 3 where β γ δ C Rα > 0. Further it is an entire unction o order Reβ 1 see Prabhakar 1971 p. 7. Shukla and Prajapati 2007 see also Srivastava & Tomovski 2009 deined and investigated the unction E γq z as α β E γ q α β z n0 γ qn z n Γαn + β n! where α β γ δ C Rα > 0 Rβ > 0 Rγ > 0 q 0 1UN and γ qn Γγ+qn denotes the Γγ generalized Pochhammer symbol which in particular reduces to q qn q γ + r 1 q r1 It is remarked that certain much more general unctions o the Mittag-Leler type have already been investigated in the literature rather systematically and extensively but or the purpose o this paper we use the unction given by 4 only. The generalized Wright unction p ψ q z deined or z C a i b j C and A i B j RA i B j 0; i 1 2 p; j 1 2 q is given by the series p ψ q z p ψ q. n a i A i 1 p z b j B j 1 q p Γa + A i1 i ik zk q Γb + B j1 j jk k! where Γz is the Euler gamma unction and the unction 5 was introduced by Wright 1935 and is known as generalized Wright unction or all values o the argument z under the condition: 4 5 Page 2 o 11

3 Mishra et al. Cogent Mathematics : q p B j A i > 1. j1 i1 6 For detailed study o various properties generalization and application o Wright unction and generalized Wright unction we reer to paper or instance see Wright The Srivastava polynomials deined by Srivastava 1968 p. 1 Equation 1 in the ollowing manner: w u S u x w w u.s A w.s x s w where u is an arbitrary positive integer and the coeicients A w.s w s 0 are arbitrary constants real or complex. On account o success o the Saigo operators Saigo in their study on various unction spaces and their application in the integral equation and dierential equations Saigo and Maeda 1998 introduced the ollowing generalized ractional and dierential operators o any complex order with Appell unction F 3 in the kernel as ollows: Let α α β β γ C and x > 0 then the generalized ractional calculus operators the Marichev- Saigo-Maeda operators involving the Appell unction or Horn s F 3 -unction are deined by the ollowing equations: x xα Γγ x x t γ1 t α 0 F 3 α α β β ; γ;1 t x 1 x t k d x I α α β+k β γ+k dx x xα Γγ x Rγ 0; k Rγ + 1 ; t x γ1 t α x F 3 α α β β ; γ;1 x t 1 t x x d k dx I α α β β +k γ+k Rγ 0; k Rγ + 1 ; x tdt Rγ > 0 tdt Rγ > and D α α ββ γ x I α αβ βγ k d I α α β +k βγ+k dx x x Rγ > 0; k Rγ + 1 ; x x I α α β β γ d dx k I α α β β+k γ+k x Rγ > 0; k Rγ Page 3 o 11

4 Mishra et al. Cogent Mathematics : For the deinition o the Appell unction F 3 the interested reader may reer to the monograph by Srivastava and Karlsson 1985 see Erdélyi Magnus Oberhettinger and Tricomi 1953 Prudnikov Brychkov and Marichev 1992 and Samko Kilbas and Marichev Following Saigo and Maeda 1998 the image ormulas or a power unction under operators 8 and 10 are given by: x ρ1 x x ραα +γ1 Γ ρ ρ + γ α α β ρ + β α ρ + β ρ + γ α α ρ + γ α β where Rρ > max { 0 Rα + α + β γ Rα β } and Rγ > 0. I α α ββ γ x ρ1 x x ρ+γαα 1 Γ 1 ρ γ + α + α Γ1 ρ + α + β γγ1 ρ β Γ1 ργ1 ρ + α + α + β γγ1 ρ + α β where Rγ > 0 Rρ < 1 + min { Rβ R α + α γ R α + β γ }. Here we used the symbol Γ representing the raction o many Gamma unctions. The computations o ractional integrals and ractional derivatives o special unctions o one and more variables are important rom the point o view o the useulness o these results in the evaluation o generalized integrals and generalized derivatives and the solution o dierential and integral equations or example see Baleanu Kumar and Purohit 2016 Kumar Purohit and Choi 2016 Nisar Purohit Abouzaid Qurashi and Baleanu 2016 Purohit Kalla and Suthar 2011 Purohit Suthar and Kalla 2012 Srivastava Suthar Parmar and Purohit 2017 Tomovski Hiler and Srivastava 2010 Tomovski Pogány and Srivastava Motivated by these avenues o applications here we establish our image ormulas or the generalized Mittag-Leler unction 4 involving let- and right-sided operators o Marichev-Saigo-Meada ractional integral operators and ractional derivatives in term o the generalized Wright unction. 2. Main results Throughout this paper we assume that a α α β β γ δ ρ μ η C >0 such that Rδ > 0 Rμ > 0 Rη > 0 q 0 1 N. Further let the constants satisy the condition a i b j C and A i B j RA i B j 0;i 1 2 p; j 1 2 q such that the condition 6 is also satisied Let-sided generalized ractional integration o product o polynomial and generalized Mittag-Leler unction In this section we establish image ormulas or the product o Srivastava polynomial and generalized Mittag-Leler unction involving let-sided operators o Marichev-Saigo-Meada ractional integral operators 8 in term o the generalized Wright unction. These ormulas are given by the ollowing theorems: Theorem 2.1 Let Rγ > 0 R > 0 Rρ > max 0 Rα + α + β γ Rα β then the generalized ractional integration o the product o generalized Mittag-Leler unction and δ μ S m n. is given by Page 4 o 11

5 Mishra et al. Cogent Mathematics : t ρ1 S m n A n s σx s 4 ψ 4 n m σ t +γ1 n x xραα m s Γη ρ + γ α α β + s ρ + β α + s ρ + s η q ρ + γ α β + s ρ + γ α α + s ρ + β + s μ δ ax. 16 Proo On using 4 and 7 writing the unction in the series orm the let-hand side o 16 leads to t ρ1 S m n σt δ μ n m n at ms x η qk A n s σ t s Γ μ + δ k k! at k I α α β β γ t ραα +γ1 x 17 Now upon using the image ormula 14 which is valid under the conditions stated with Theorem 2.1 we get t ρ1 S m n σt δ μ at x n m n m s A n s σx s x ραα +γ1 Γρ + γ α α β + s + k Γη Γρ + γ α β + s + k Γρ + β α + s + k Γρ + s + k Γη + qk ax k Γρ + γ α α + s + k Γρ + β + s + k Γμ + δk k! 18 Interpreting the right-hand side o the above equation in view o the deinition 5 we arrive at the result 16. On setting n 0 A 0 0 x 1 in 16 we obtained the ollowing particular case o Theorem 2.1: Corollary 2.1 Let the conditions o Theorem 2.1 are satisied then the ollowing ormula holds ture I α α ββ γ t ρ1 +γ1 x xραα Γη 4 ψ 4 ρ + γ α α β ρ + β α ρ η q ρ + γ α β ρ + γ α α ρ + β μ δ ax. 19 Remark 1 I we set q 1 in Corollary 2.1 we arrive at the known result given by Chouhan Khan and Saraswat 2014 Equation 13. Now we present some special cases o 19 as below: For α α + β α β 0 β τ γ α we obtain the ollowing relationship x I x where the operator I deined by 20 denotes the Saigo ractional integral operator Saigo 1978 which is x I x xατ x t α1 Γα 2 F 1 α + τ η; α;1 tx tdt Rα > Corollary 2.2 Let Rγ > 0 Rυ > 0 Rρ > max 0 Rβ τ then there hold the ollowing ormula: Page 5 o 11

6 Mishra et al. Cogent Mathematics : I t ρ1 S m n σt n m n x xρβ1 ms A Γη ns σx s 3 ψ 3 ρ β + τ + s ρ + s η q ρ + α + τ + s ρ β + s μ δ ax. 22 Remark 2 I we set q 1 τ γ and n 0 A 0 0 x 1 in Corollary 2.2 we arrive at the known result given by Ahmed 2014 Equation Right-sided generalized ractional integration o product o polynomial and generalized Mittag-Leler unction In this part we establish image ormulas or the product o Srivastava polynomial and generalized Mittag-Leler unction involving right-sided operators o Marichev-Saigo-Meada ractional integral operators 10 in term o the generalized Wright unction. These ormulas are given by the ollowing theorems: Theorem 2.2 For Rγ > 0 R1 γ ρ < 1 + min Rβ Rα + α γ Rα + β γ we have t γρ S m n A n s σ x s 4 ψ 4 n m n m s σ t x xραα Γη 23 α + α + ρ s α + β + ρ s ρ β + γ s η q μ δ α + α + β + ρ s α β + ρ + γ s ρ + γ s ax. Proo On using 4 and 7 the let-hand side o 23 can be written as: I αα β β γ t γρ S m n σt δ μ n m n at m s x A n s σt s η qk Γμ + δ k at k t ραα x 24 which on using the image ormula 15 arrive at n m t γρ S m n σt δ μ at x n m s A σx s x ραα n s Γη Γα + α + ρ s + k Γα + α + β + ρ s + k ax k k! Γα + β + ρ s + k Γρ β + γ s + kγη + qk Γα β + ρ + γ s + k Γρ + γ s + k Γμ + δk 25 Interpreting the right-hand side o the above equation in view o the deinition 5 we arrive at the result 23. On setting n 0 A 0 0 x 1 in 23 we obtained the ollowing particular case o Theorem 2.2. Corollary 2.3 The generalized ractional integration o generalized Mittag-Leler unction is δ μ given by t γρ x xραα Γη 4 ψ 4 α + α + ρ α + β + ρ ρ β + γ η q μ δ α + α + β + ρ α β + ρ + γ ρ + γ ax Page 6 o 11

7 Mishra et al. Cogent Mathematics : provided Rγ > 0 R1 γ ρ < 1 + min Rβ Rα + α γ Rα + β γ. Remark 3 I we set q 1 in Corollary 2.3 we arrive at the known result given by Chouhan et al Equation 15. When we let α α + β α β 0 β τ γ α then we obtain the relationship x I x where the Saigo ractional integral operator Saigo 1978 is deined by 26 I x 1 t x α1 t αβ F Γα 2 1 α + β τ; α;1 xt tdt. x 27 Corollary 2.4 I Rα > 0 R > 0 R1 γ ρ < 1 + min Rβ Rτ then we have I t γρ S m n A n s σ x s 3 ψ 3 n m σ t n x xραβ m s Γη α + β + ρ s ρ + τ + α s η q μ δ 2α + β + τ + ρ s ρ + α s ax. 28 Remark 4 I we set q 1 τ γ and n 0 A 0 0 x 1 in Corollary 2.4 we arrive at the known result given by Ahmed 2014 Equation Let-sided generalized ractional dierentiation o product o polynomial and generalized Mittag-Leler unction Now we shall establish image ormulas or the product o Srivastava polynomial and generalized Mittag-Leler unction involving let-sided operators o Marichev-Saigo-Meada ractional dierentiation operators 12 in term o the generalized Wright unction. These ormulas are given by the ollowing theorems: Theorem 2.3 The generalized ractional dierentiation o the product o generalized Mittag- Leler unction and Srivastava polynomials δ μ Sm n is given by t ρ1 S m n A n s σ x s 4 ψ 4 n m n ms σt γ1 x xρ+α+α Γη 29 ρ γ + α + α + β + s ρ β + α + s ρ + s η q ρ γ + α + β + s ρ γ + α + α + s ρ β + s μ δ ax where Rγ > 0 R > 0 Rρ > max 0 Rγ α α β Rβ α. Proo On using 4 and 7 writing the unction in the series orm the let-hand side o 29 leads to t ρ1 S m n σ t δ μ n m at x η qk Γ μ + δ k at k I α α β β γ 0+ k! n m s A n s σ t s t ραα +γ1 x 30 Now upon using the image ormula 14 which is valid under the conditions stated with Theorem 2.3 we get Page 7 o 11

8 Mishra et al. Cogent Mathematics : t ρ1 S m n σ t δ μ n m n at m s x A n s σ x s x ραα +γ1 Γρ γ + α + α + β + s + k Γη Γρ + γ + α + β + s + k Γρ β + α + s + k Γρ + s + kγη + qk ax k Γρ + γ + α + α + s + kγρ β + s + k Γμ + δk k! 31 Interpreting the right-hand side o the above equation in view o the deinition 5 we arrive at the result 29. On setting n 0 A 0 0 x 1 in 29 we obtained the ollowing particular case o Theorem 2.3. Corollary 2.5 Under the conditions Rγ > 0 R > 0 and Rρ > max 0 Rγ α α β Rβ α the ollowing ormula holds t ρ1 γ1 x xρ+α+α Γη 4 ψ 4 ρ γ + α + α + β ρ β + α ρ η q ρ γ + α + β ρ γ + α + α ρ β μ δ ax. 32 Now we present one more special case o 29 by making use o identity 20 as given below: Corollary 2.6 The ollowing generalized ractional dierentiation ormula holds D n m t ρ1 S m n σ t γ1 x xρ+α+α Γη n m s A n s σ x s ψ 3 3 ρ + α + β + τ + s ρ + s η q ρ + β + s ρ + τ + s μ δ ax 33 where Rγ > 0 Rυ > 0 and Rρ > max 0 Rβ τ. Remark 5 I we set q 1 τ γ and n 0 A 0 0 x 1 in Corollary 2.6 we arrive at the known result given by Ahmed 2014 Equation Right-sided generalized ractional dierentiation o product o polynomial and generalized Mittag-Leler unction Here we establish image ormulas or the product o Srivastava polynomials and generalized Mittag- Leler unction involving right-sided operators o Marichev-Saigo-Meada ractional dierentiation operators 13 in term o the generalized Wright unction. These results are given as ollows: Theorem 2.4 I Rγ > 0 R1 γ ρ < 1 + min Rβ Rα + α γrα + β γ then we have t γρ S m n A n s σ x s 4 ψ 4 n m σ t n x xρ+α+α m s Γη ρ α α s ρ β α s ρ + β γ s η q μ δ ρ α α β s ρ α + β γ s ρ γ s ax. 34 Page 8 o 11

9 Mishra et al. Cogent Mathematics : Proo By using 4 and 7 the let-hand side o 34 can be written as t γρ S m n σ t δ μ n m at x η qk Γμ + δ k at k I α α β β γ n m s A n s σ t s t ρ+α+α x 35 which on using the image ormula 15 arrive at n m t γρ S m n σ t δ μ at x n m s A σx s x ρ+α+α n s Γη Γρ α α s + k Γρ α α β s + k ax k k! Γρ α β s + kγρ + β γ s + kγη + qk Γρ α + β γ s + kγρ γ s + k Γμ + δ k 36 Interpreting the right-hand side o the above equation in view o the deinition 5 we arrive at the result 34. Further on setting n 0 A 0 0 x 1 in 34 we obtained the ollowing particular case o Theorem 2.4. Corollary 2.7 Let the conditions o Theorem 2.4 are satisied then the ollowing ormula holds t γρ δμ at x xρ+α+α Γη 4 ψ 4 ρ α α s ρ β α ρ + β γ η q μ δ ρ α α βs ρ α + β γs ρ γ ax. 37 Now by using the identity 26 we present certain special cases o 34 as given below: Corollary 2.8 The generalized ractional dierentiation ormula associated with the product o generalized Mittag-Leler unction and Srivastava polynomials is given by D t γρ S m n σ t x xρ+α+α Γη n m n m s A n s σ x s ρ α β s ρ + τ s η q ψ 3 3 μ δ ρ α β s ρ α s ax 38 provided Rγ > 0 R > 0 R1 γ ρ < 1 + min Rβ Rτ. Remark 6 Finally i we set q 1 τ γ and n 0 A hence S m 0 x 1 in Corollary 2.8 we arrive at the known result given by Ahmed 2014 Equation 6.1. Funding The authors received no direct unding or this research. Author details Vishnu Narayan Mishra 1 vishnunarayanmishra@gmail.com ORCID ID: D.L. Suthar 2 dlsuthar@gmail.com S.D. Purohit 3 sunil_a_purohit@yahoo.com ORCID ID: 1 Department o Applied Mathematics & Humanities Sardar Vallabhbhai National Institute o Technology Surat Gujarat India. 2 Department o Mathematics Wollo University Dessie Campus P.O. Box South Wollo Amhara Region Ethiopia. 3 Department o HEAS Mathematics Rajasthan Technical University Kota India. Citation inormation Cite this article as: Marichev-Saigo-Maeda ractional calculus operators Srivastava polynomials and generalized Mittag-Leler unction Vishnu Narayan Mishra D.L. Suthar & S.D. Purohit Cogent Mathematics : Page 9 o 11

10 Mishra et al. Cogent Mathematics : Reerences Ahmed S On the generalized ractional integrals o the generalized Mittag-Leler unction. SpringerPlus Baleanu D. Kumar D. & Purohit S. D Generalized ractional integrals o product o two H-unctions and a general class o polynomials. International Journal o Computer Mathematics Chouhan A. Khan A. M. & Saraswat S A note on Marichev-Saigo-Maeda ractional integral operator. Journal o Fractional Calculus and Applications Erdélyi A. Magnus W. Oberhettinger F. & Tricomi F. G Higher transcendental unctions Vol. 1. New York NY: McGraw-Hill Book Company. Kilbas A. A. Srivastava H. M. & Trujillo J. J Theory and Applications o Fractional Dierential Equations North- Holland Mathematical Studies Vol Amsterdam: Elsevier North-Holland Science Publishers. Kiryakova V Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series Vol Harlow: Longman Scientiic & Technical.Harlow: copublished in the United States with John Wiley & Sons New York Kumar D. Purohit S. D. & Choi J Generalized ractional integrals involving product o multivariable H-unction and a general class o polynomials. The Journal o Nonlinear Science and its Applications Miller K. S. & Ross B An introduction to the ractional calculus and ractional dierential equations A Wiley- Interscience Publication. New York NY: John Wiley & Sons. Mishra L. N. & Agarwal R. P On existence theorems or some nonlinear unctional-integral equations. Dynamic Systems and Applications Mishra L. N. Agarwal R. P. & Sen M Solvability and asymptotic behavior or some nonlinear quadratic integral equation involving Erdélyi-Kober ractional integrals on the unbounded interval. Progress in Fractional Dierentiation and Applications Mishra L. N. & Sen M On the concept o existence and local attractivity o solutions or some quadratic Volterra integral equation o ractional order. Applied Mathematics and Computation Mishra L. N. Srivastava H. M. & Sen M On existence results or some nonlinear unctional-integral equations in Banach algebra with applications. International Journal o Analysis and Applications Mittag-Leler G. M Sur la nuovelle unction Eαx. Comptes Rendus de l Académie des Sciences Paris Nisar K. S. Purohit S. D. Abouzaid M. Qurashi M. A. & Baleanu D Generalized k-mittag Leler unction and its composition with Pathway integral operators. The Journal o Nonlinear Science and its Applications Prabhakar T. R A singular integral equation with a generalized Mittag Leler unction in the kernel. Yokohama Mathematical Journal Prudnikov A. P. Brychkov Y. A & Marichev O. I Integrals and series Vol. Inverse Laplace transorms. New York NY: Gordon and Breach Science Publishers. Purohit S. D Solutions o ractional partial dierential equations o quantum mechanics. The Advances in Applied Mathematics and Mechanics Purohit S. D. & Kalla S. L On ractional partial dierential equations related to quantum mechanics. Journal o Physics A: Mathematical and Theoretical 44 8 pp. Purohit S. D. Kalla S. L. & Suthar D. L Fractional integral operators and the multiindex Mittag-Leler unctions. Scientia Series A : Mathematical Sciences N.S Purohit S. D. Suthar D. L. & Kalla S. L Marichev-Saigo- Maeda ractional integration operators o the Bessel unctions. Matematiche Catania Saigo M A remark on integral operators involving the Gauss hypergeometric unctions. Mathematical Reports o Kyushu University Saigo M A certain boundary value problem or the Euler-Darboux equation. Mathematicae Japonicae Saigo M. & Maeda N More generalization o ractional calculus Transorm methods & special unctions Varna 96 pp Soia: Bulgarian Academy o Sciences. Samko S. Kilbas A. & Marichev O Fractional integrals and derivatives: theory and applications. Yverdon: Gordon and Breach Science Publishers. Shukla A. K. & Prajapati J. C On a generalization o Mittag-Leler unction and its properties. Journal o Mathematical Analysis and Applications Srivastava H. M On an extension o the Mittag-Leler unction. Yokohama Mathematical Journal Srivastava H. M A contour integral involving Fox s H-unction. Indian Journal o Mathematics Srivastava H. M Some amilies o Mittag-Leler type unctions and associated operators o ractional calculus. TWMS Journal o Pure and Applied Mathematics Srivastava H. M. & Karlsson P. W Multiple Gaussion hypergeometric series Ellis Horwood series: Mathematics and its applications. Chichester: Ellis Horwood. Halsted Press John Wiley & Sons Inc. New York Srivastava H. M. & Saigo M Multiplication o ractional calculus operators and boundary value problems involving the Euler-Darboux equation. Journal o Mathematical Analysis and Applications Srivastava H. M. & Saxena R. K Operators o ractional integration and their applications. Applied Mathematics and Computation Srivastava H. M. & Tomovski Z Fractional calculus with an integral operator containing a generalized Mittag- Leler unction in the kernel. Applied Mathematics and Computation Suthar D. L. Parmar R. K. & Purohit S. D Fractional calculus with complex order and generalized hypergeometric unctions. Nonlinear Science Letters A Tomovski Ž. Hiler R. & Srivastava H. M Fractional and operational calculus with generalized ractional derivative operators and Mittag-Leler type unctions. Integral Transorms and Special Functions Tomovski Ž. Pogány T. K. & Srivastava H. M Laplace type integral expressions or a certain three-parameter amily o generalized Mittag-Leler unctions with applications involving complete monotonicity. Journal o The Franklin Institute Wiman A Über den Fundamentalsatz in der Teorie der Funktionen Eαx On the undamentals in the theory o the unctions Eαx. Acta Mathematica Wright E. M The asymptotic expansion o the generalized hypergeometric unctions. Journal o the London Mathematical Society Wright E. M The asymptotic expansion o integral unctions deined by Taylor series. Philosophical Transactions o the Royal Society London A Wright E. M The asymptotic expansion o the generalized hypergeometric unction II. Proceedings o the London Mathematical Society Page 10 o 11

11 Mishra et al. Cogent Mathematics : The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY 4.0 license. You are ree to: Share copy and redistribute the material in any medium or ormat Adapt remix transorm and build upon the material or any purpose even commercially. The licensor cannot revoke these reedoms as long as you ollow the license terms. Under the ollowing terms: Attribution You must give appropriate credit provide a link to the license and indicate i changes were made. You may do so in any reasonable manner but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others rom doing anything the license permits. Cogent Mathematics ISSN: is published by Cogent OA part o Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics or your article Rapid online publication Input rom and dialog with expert editors and editorial boards Retention o ull copyright o your article Guaranteed legacy preservation o your article Discounts and waivers or authors in developing regions Submit your manuscript to a Cogent OA journal at Page 11 o 11