Weighted hypergeometric functions and fractional derivative

Transcription

1 Restrepo et al. Advances in Difference Equations 217) 217:15 DOI /s R E S E A R C H Open Access Weighted hypergeometric functions and fractional derivative JE Restrepo 1,AKılıçman 2*,PAgarwal 3 and Omer Altun 2 * Correspondence: akilic@upm.edu.my 2 Department of Mathematics, University Putra Malaysia, Serdang, Selangor 434 UPM, Malaysia Full list of author information is available at the end of the article Abstract We introduce some weighted hypergeometric functions and the suitable generaliation of the Caputo fractional derivation. For these hypergeometric functions, some linear and bilinear relations are obtained by means of the mentioned derivation operator. Then some of the considered hypergeometric functions are determined in terms of the generalied Mittag-Leffler function E γ j),l j ) ρ j ),λ [ 1,..., r ] Mainardi in Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 21) and the generalied polynomials S m n [x] Srivastava in Indian J. Math. 14:1-6, 1972). The boundary behavior of some other class of weighted hypergeometric functions is described in terms of Frostman s α-capacity. Finally, an application is given using our fractional operator in the problem of fractional calculus of variations. MSC: 3A1; 33E12; 26D7 Keywords: weighted Caputo fractional derivative; weighted hypergeometric function; generating function; Srivastava polynomials; generalied Mittag-Leffler function; α-capacity 1 Introduction Recently, many specialists have investigated special functions due to the important role of these functions in mathematical, physical and engineering problems. Various extensions of some special functions were studied in many works see, e.g.,[3 6]). For introducing some new weighted hypergeometric functions, we use the weighted extension of Euler s beta-function a particular case associated with the extensions considered in [7]): B ω α,β) x, y) 1 t x 1 1 t) y 1 ω α,β) t, u, v) dt, 1.1) where Re x >,Re y >andα, β,, u, v are real or complex parameters and ω α,β) t, u, v)is afunctionoftheclass, i.e. such that the integral 1.1) is absolutely convergent. Besides, by writing B ω x, y) we mean that this function and the function ω do not depend on α and β.onecanseethat,ifωt, p,)e p t) with Re p >,thenb ω x, y)minre x, Re y} >) becomes the extension of Euler s beta-function considered by Chaudhry et al. [8]. Besides, The Authors) 217. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 2 of 11 by a straightforward calculation, it follows in the general case that B ω x, y +1)+B ω x +1,y)B ω x, y), ω. In the last years, the interest to application of the fractional derivative operators a considerable growth. Numerous publications have been devoted to the solutions of different problems by applying these operators see [9 11]). Recall that the well-known Caputo fractional derivative is defined as D μ f ): 1 Ɣm μ) t) m dm f t) dt, dtm where m 1<Re μ < m, m N the set of positive integers). Here and in the following, let C, R, N, andz be the sets of complex numbers, real numbers, positive integers, and non-positive integers, respectively. We introduce the general Caputo fractional derivative suitable with the function 1.1) by the following generaliation of that defined in [12, 13]: D μ,τ 1 f ): Ɣm μ) t) m τt, u, v) dm f t) dt, 1.2) dtm where m 1<Re μ < m, m N and u, v are real or complex parameters and τ is assumed to be a function of the class, i.e. such that the integral 1.2)isabsolutelyconvergent.Itis noted that 1.2) becomes the extended Caputo fractional derivative [12] whenτt, p, ) e p2 t t) Re p > ), while it becomes the classical Caputo fractional derivative when τ 1. 2 Weighted hypergeometric functions Inthesameformasdefinedin[12],weintroducesomeweightedversionsoftheGauss hypergeometric function 2 F 1, the Appell hypergeometric functions F 1, F 2 see [14]), the Lauricella hypergeometric function FD;ω 3, the generalied Gauss hypergeometric function F ω and the generalied confluent hypergeometric function 1 F1 ω. Everywhere in this paper we assume that m N, ω and Bx, y)minre x, Re y} >)istheclassicalbetafunction. Definition 2.1 The ω-weighted extended Gauss hypergeometric function is 2F 1 a, b; c; ; ω): n a) n b) n B ω b m + n, c b + m) n b m) n Bb m, c b + m), 2.1) where <1andm < Re b < Re c. For an illustration of the Gauss hypergeometric function see [15, 16]. Definition 2.2 The ω-weighted extended Appell hypergeometric function F 1 is F 1 a, b, c; d; x, y; ω): n,k where x <1, y <1andm < Re a < Re d. a) n+k b) n c) k B ω a m + n + k, d a + m) x n y k a m) n+k Ba m, d a + m) k!, 2.2)

3 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 3 of 11 Definition 2.3 The ω-weighted extended Appell hypergeometric function F 2 is F 2 a, b, c; d, e; x, y; ω) a) n+k b) n c) k B ω b m + n, d b + m) : b m) n c m) k Bb m, d b + m) n,k B ωc m + k, e c + m) x n y k Bc m, e c + m) k!, 2.3) where x + y <1,m < Re b < Re d and m < Re c < Re e. Besides, we just consider one of the form of Appell hypergeometric function [12, 17]. Definition 2.4 The ω-weighted extended Appell hypergeometric function F 3 D,ω is FD,ω 3 a, b, c, d; e; x, y, ; ω) a) n+k+r b) n c) k d) r B ω a m + n + k + r, e a + m) x n y k r : a m) n+k+r Ba m, e a + m) k! r!, 2.4) n,k,r where x + y + <1andm < Re a < Re e. Note that the functions defined above become those in [12] whenωt, p,)e p t) and Re p >. Besides, for ω 1, these functions reduce to the well-known Gauss hypergeometric function 2 F 1,AppellfunctionsF 1, F 2 and Lauricella function FD 3,respectively. Definition 2.5 The ω-weighted generalied Gauss hypergeometric function F ω is F ω a, b; c; ) n a) n B ω b + n, c b) Bb, c b) where <1andRe c > Re b >. n, 2.5) Definition 2.6 The ω-weighted generalied confluent hypergeometric function 1 F ω 1 is 1F ω 1 b; c; ) n where <1andRe c > Re b >. B ω b + n, c b) n Bb, c b), 2.6) For some examples of functions F ω and 1 F ω 1,see[7]. Remark 2.1 If ω α,β) p t, p,) 1 F 1 α; β; )whereminre x, Re y, Re α, Re β} >and t) Re p, then B ω α,β) x, y) is the function B p α,β) x, y) definedin[5] seep.32in[5], also, p.1748 in [18]). Hence, F ω a, b; c; )F p α,β) a, b; c; )and 1 F1 ωb; c; ) 1F α,β,p) 1 b; c; )arethe same as those in [18], pp see also [5], p.39). The next definition, it will be useful to introduce a general result.

4 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 4 of 11 Definition 2.7 Let f ) : n a n n and g) : n b n n be two power series whose convergence or radii are R f and R g, respectively. Then the Hadamard product of f ) and g) isthepowerseriessee[18, 19]) f g)): a n b n n, n whose convergence radius R satisfies the inequality R f R g R. Remark 2.2 The above definitions can be considered with B α,β) ω instead of B ω. 3 The weighted Caputo derivative Below, we establish some useful statements, where we apply the weighted Caputo fractional derivative. The next results are some generaliations of those in [12, 14, 18] and some others. Hence, the tools to prove them lemmas and theorems) are similar. Lemma 3.1 If m 1<Re μ < m, ω, ω, s, w C and Re μ < Re λ, then D μ,ω [ λ ] Ɣλ +1)B ωλ m +1,m μ) Ɣλ μ +1)Bλ m +1,m μ) λ μ. Proof Indeed, D μ,ω [ λ ] 1 Ɣm μ) Ɣλ +1) Ɣm μ)ɣλ m +1) Ɣλ +1) λ μ Ɣm μ)ɣλ m +1) t) m ωt, s, w) dm dt m tλ dt 1 Ɣλ +1)B ωλ m +1,m μ) Ɣλ μ +1)Bλ m +1,m μ) λ μ. t) m t λ m ωt, s, w) dt 1 u) m u λ m ωu, s, w) du Theorem 3.1 If f ) n a n n is an analytic function in the disk < ρ, ω and ω, then D μ,ω f ) } n D μ,ω n }, where m 1<Re μ < m. Proof As the power series converges uniformly, the integral of D μ,ω f )} converges absolutely, and hence the desired result follows by a straightforward calculation. Theorem 3.2 Let m 1<Re λ μ < m < Re λ, κ C, ω and ω. Then λ 1 1 ) κ} Ɣμ) 2 F 1 κ, λ; μ; ; ω), <1. 3.1)

5 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 5 of 11 Proof Using the power series of 1 ) κ,theorem3.1, Lemma 3.1 and equation 2.1), we obtain λ 1 1 ) κ} n κ) n n Ɣμ) λ 1 n } n κ) n λ+n 1 } κ) n Ɣλ + n)b ω λ m + n, m λ + μ) n Ɣμ + n)bλ m + n, m λ + μ) n Ɣμ) 2 F 1 κ, λ; μ;, ω). κ) n λ) n B ω λ m + n, m λ + μ) n λ m) n Bλ m, μ λ + m) Theorem 3.3 If m 1<Reλ μ)<m < Re λ, ω and ω, then λ 1 1 r) κ 1 s) θ} Ɣμ) F 1λ, κ, θ; μ; r; s; ω), 3.2) for r, s, κ, θ C, r <1and s <1. Proof Using the power series of 1 r) κ,1 s) θ,theorem3.1, Lemma 3.1 and 2.2), we get λ 1 1 r) κ 1 s) θ} ) κ) n θ) k r n s k λ+n+k 1 k! n,k n,k κ) n θ) k r n s k λ+n+k 1 } k! κ) n θ) k r n s k k! n,k Ɣμ) n,k Ɣμ) F 1λ, κ, θ; μ; r; s; ω). Ɣλ + n + k)b ω λ m + n + k, m λ + μ) Ɣμ + n + k)bλ m + n + k, m λ + μ) n+k λ) n+k κ) n θ) k B ω λ m + n + k, m λ + μ) r) n s) k λ m) n+k Bλ m, m λ + μ) k! Theorem 3.4 If m 1<Reλ μ)<m < Re λ, ω, ω and m < Re β < Re γ, then λ 1 1 ) α2f )) x 1 α, β; γ ; 1 ; ω Ɣμ) F 2 α, β, λ, γ ; μ; x, ; ω), x + < )

6 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 6 of 11 Proof By the power series of 1 a) α,2.1)and3.1), we obtain λ 1 1 ) α2f )) x 1 α, β; γ ; 1 ; ω λ 1 1 ) α n} Ɣμ) n,k n β) n B ω β m + n, γ β + m) x n β m) n Bβ m, γ β + m) +k β) n λ) k B ω β m + n, γ β + m) β m) n λ m) k Bβ m, γ β + m) B ωλ m + k, μ λ + m) x n k Bλ m, μ λ + m) k! Ɣμ) F 2λ, α, β; μ; a; b; ω). 4 Generating functions Below we obtain some bilinear generating relations for the weighted extended hypergeometric function 2 F 1. In a sense, these relations are similar to those in [7, 12] by taking some particular weights ω belonging to. Theorem 4.1 If m 1<Reλ μ)<re μ, ω and ω, then n ) 2F 1 α + n, λ; μ; ; ω)t n ) α 2F 1 α, λ; μ; ; ω, 4.1) where < min1, 1 t }. Proof Note that by the identity of [12] [ ] α 1 ) t ) α 1 ) α. We write a power series in the left hand side n ) t n 1 ) α ) α 1 ) α, 1 where t < 1. Besides, by multiplying both sides by λ 1 and applying our operator,weget } t n λ 1 1 ) α n n 1 t) α λ 1 1 ) α }. As t < 1 and < Re μ < Re λ, the fractional derivative can be replaced inside the sum: n λ 1 1 ) α n} t n ) α λ 1 1 ) α }. Finally, by Theorem 3.2,weget4.1).

7 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 7 of 11 Theorem 4.2 If m 1<Reλ μ)<re μ, ω and ω, then n where t < ) 2F 1 β n, λ; μ; ; ω)t n ) α t F 1 β, α, λ; μ; ; ; ω, 4.2) Proof By the identity of [12]weget [ ] α 1 1 )t ) α 1+ t ) α, and we write the power series in the left hand side n 1 ) n t n ) α 1 t ) α, t <1/ 1. Besides, by multiplying both sides by λ 1 1 ) β and applying the weighted Caputo fractional derivative we get n } λ 1 1 ) β n t n 1 t) α λ 1 1 ) β 1 t ) α }. As t < and < Re μ < Re λ, the derivative can be replaced inside the sum: n λ 1 1 ) α n} t n ) α λ 1 1 ) β 1 t ) α }. Hence, equation 4.2)followsby Theorems3.2 and 3.3. Theorem 4.3 If m 1<Reβ γ )<Re β, m < Re λ < Re μ, ω and ω, then n ) ut 2F 1 α + n, λ; μ; ; ω) 2 F 1 n, β; γ ; u; ω)f 2 α, λ, β; μ, γ ;, ; ω. 4.3) Proof If t tends to 1 u)t in equation 4.1) and multiplying both sides by u β 1,weobtain 2F 1 α + n, λ; μ; ; ω)u β 1 1 u) n t n u β 1[ 1 1 u)t ] ) α 2F 1 α, λ; μ; 1 1 u)t ; ω. n Hence, applying the fractional derivative D β λ,ω u n D β λ,ω u 2F 1 α + n, λ; μ; ; ω)du β λ,ω u β 1 1 u) n} t n u β 1[ 1 1 u)t ] α 2F 1 α, λ; μ; to both sides we get )} 1 1 u)t ; ω,

8 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 8 of 11 where <1, 1 u t <1and 1 + ut < 1. This formula is the same as n D β λ,ω u 2F 1 α + n, λ; μ; ; ω)du β λ,ω u β 1 1 u) n} t n [ u β 1 1 ut ] α2f 1 α, λ; μ; )} 1 ut ; ω. Therefore, we obtain the desired statement 4.3) by Theorems 3.2 and 3.4. Now, using the weighted function F ω, we introduce a generaliation of the generating relation given in [18], pp see also [14]). Theorem 4.4 If Re c > Re b >and ω, then 1 + t) λ ) F ω a, b; c, /1 + t) r 1) r λ) r F ω a, b; c; ) 1 F 1 λ + r; λ; ) tr,, λ C, t <1. r! Proof TheproofrunsparalleltothatofTheorem2.1in[18]. We omit the details. Remark 4.1 By Remark 2.2, we can get all results of the sections the weighted Caputo derivative and Generating functions), in the same way. Besides, these results become those in [12] when we consider the particular ωt, p,)e p t) and Re p >. 5 Further results and observations Note that, if ω α,β) Ɣβ) t, p, s) Ɣα)Ɣβ α) Ɣ pt2s), then Bα,β) ω becomes the following generalied gamma function see [5], pp.32-33): B ω α,β) Ɣβ) α s, β α) Ɣα)Ɣβ α) 1 t α s 1 1 t) β α 1 Ɣ pt 2s) dt Ɣ p α,β) s), where minre y, Re α, Re β, Re p, Re s} >,Ɣ p s) is Chaudhry s gamma function [8] and Ɣ p α,β) s)isdefinedin[5], p.33. Now, we shall write some of the considered weighted functions in terms of the wellknown Mittag-Leffler function [1]. We work with the generaliation of the multivariable Mittag-Leffler function E γ j),l j ) ρ j ),λ [ 1,..., r ]introducedbysaxenaet al. [2], p.547, Eq. 7.1) see also [21], pp.2-3) and the generalied polynomials Sn m [x]see[2], p.1, Eq. 1)). Corollary 5.1 If ωx 1, 1,)S m 1 n 1 [x λ x 1) η ] exp 1 x 1 ) μ 11 x 1 ) δ 1} and η, λ 1, δ 1, μ 1 C; j, Rea m + n + k + λ 1 j + μ 1 k 1 )>;Red a + m + nj + δ 1 k 1 )>,then F 1 a, b, c; d; x, y; ω) n,k [n 1 /m 1 ] j a) n+k b) n c) k Ɣd)Ɣa m + n + k + λ 1 j)ɣd a + m + λj) a m) n+k Ɣa m)ɣd a + m) n 1) m1 j A n1,je 1,a m+n+k+λ 1j,d a+m+ηj),1,μ 1,δ 1 ) 1,μ j! 1 +δ 1 )),1,d a+m+ηj+a m+n+k+λ 1 j) [ 1] xn y k k!, where x <1, y <1and m < Re a < Re d.

9 Restrepoetal. Advances in Difference Equations 217) 217:15 Page 9 of 11 Proof The desired equalityholdsbycorollary3.13in [21] p.13) and Definition 2.2. Corollary 5.2 If ωx 1, 1,)S m 1 n 1 [x λ x 1) η ]E γ 1),l 1 ) ρ 1 ),λ [ 1x 1 ) μ 11 x 1 ) δ 1] and η, λ, λ 1, γ 1, ρ 1, δ 1, μ 1 C; Re ρ 1 >;Re γ 1 >;j, Rea m + n + k + r + λ 1 j + μ 1 k 1 )>;Ree a + m + ηj + δ 1 k 1 )>;l 1 N; λ / Z, then FD,ω 3 a, b, c, d; e; x, y, ; ω) n,k,r [n 1 /m 1 ] j n 1 ) m1 j j! a) n+k+r b) n c) k d) r Ɣe)Ɣa m + n + k + r + λ 1 j) a m) n+k+r Ɣa m)ɣe a + m) Ɣe a + m + λj)a n1,je γ 1,a m+n+k+r+λ 1 j,e a+m+n 1 j),l 1,μ 1,δ 1 ) ρ 1,μ 1 +δ 1 )),λ,e a+m+n 1 j+a m+n+k+r+λ 1 j) [ 1] xn y k r k! r!, where x + y + <1and m < Re a < Re e. Proof The desired equality follows by a straightforward calculation using Corollary 3.1 in [21], p.12 and Definition 2.4. To present another application of the above results, we recall some well-known facts. We have Djrbashian s Cauchy type kernel [22], p.76: C ω ) + k k k, 1, k k 1 t k 1 ωt) dt, k N. 5.1) One can see that C ω ) is a holomorphic function in the unit disc we denote this by D C : <1}) for any ωt) where is the class defined in [22], p.76). Remark 5.1 For the particular case of power functions ωx)1 x) α, 1<α <,C ω ) is the 1+α order of the ordinary Cauchy kernel: C ω ) 1 1 ) : C α), and C 1+α 1 ) 1, D. 5.2) 1 Definition 5.1 Let E [, 2π]beaBorelmeasurablesetB-set) and ω.itissaidthat E is of positive ω-capacity C ω E) > ) if there exists a nonnegative B-measure μ supported and finite on E and such that S 1 lim max r 1 ϕ 2π 2π Cω re iϕ θ) ) dμθ)<+. If there is no such a measure, i.e. if S 1 for any nonnegative B-measure, then E is said to be of ero ω-capacity C ω E)). Note that if we take ωx)1 x) α 1 < α < ), the last definition becomes that of the well-known Frostman α-capacity [22]. Corollary 5.3 If σ θ) is a function of bounded variation on [,2π],then the function F α ) 1 2π 2F 1 α +1,1;1;e iθ ;1 ) dσ θ), 1 < α <, <1, 2π

10 Restrepoetal.Advances in Difference Equations 217) 217:15 Page 1 of 11 has non-ero, finite nontangential boundary values F α e iϕ ) at all points ϕ [, 2π], with a possible exception of a set S [, 2π] of ero α-capacity. Proof First, one can see that 2F 1 α +1,1;1;e iθ ;1 ) 1 1 e iθ ) ω α+1 1 e iθ, α ), θ [, 2π]. Besides, the function ωx) 1 x) α 1 < α < ) belongs to the class, and therefore Ce iθ ; ω) becomesce iθ ; α)ω 1 e iθ, α) θ [, 2π]) by Remark 5.1. Hence,byTheorem 2.5 in [22], p.112, we find that the function F α ) has non-ero, finite nontangential boundary values at all points e iϕ ϕ [, 2π]), except in a set S [, 2π] oferoαcapacity. On the other hand, if f is a continuous function in [, b]andt [, b], then D μ,1 t f t)the operator defined in equation 1.2)) becomes RC Dμ t f t)see[23], p.4, equation 1)), where m 1<μ < m, m N Iμ ). Thus, if we consider the fractional problem about calculus of variations described in [23, 24] with our operator D μ,τ f ), under the conditions R, m 1<μ < m and m N. We find that the functional I [ p ) ] b L t, pt), D μ,1 t pt) ) dt, where [, b] R, <b, <μ < 1, and the functions pt) and the Lagrangian L : t, p, vl) Lt, p, vl) are considered to be functions of class C 2 p ) C 2 [, b]; R), L,, ) C 2 [, b] R R; R)), satisfies the fractional Euler-Lagrange equation in the senseofries-caputosee[23, 24]), i.e. 2 L t, pt), D μ,1 t pt) ) 1 d Ɣ1 μ) dt t t θ) μ 3 L θ, pθ), D μ,1 θ pθ) )) dθ, where i L is the partial derivative of L with respect to its ith argument i 1,2,3)andp ) is an extremier of the functional I[p )] for all t [, b]. 6 Conclusions We hope to find some engineering applications related to our new results. Also, we analye the possibilities to find solutions of partial differential equations or differential equations in terms of our results. Besides, we are trying to write the weighted hypergeomeric functions like Poisson integrals considering some special weights to find boundary values, factoriations of this functions and applications. Competing interests The authors declare that they have no competing interests. Authors contributions All authors jointly worked on deriving the results and approved the final manuscript. Author details 1 Institute of Mathematics, University of Antioquia, Cl , Medellin, Colombia. 2 Department of Mathematics, University Putra Malaysia, Serdang, Selangor 434 UPM, Malaysia. 3 Department of Mathematics, Anand International College of Engineering, Jaipur, 3312, India.

11 Restrepoetal.Advances in Difference Equations 217) 217:15 Page 11 of 11 Acknowledgements The authors would like to thank the reviewers for careful reading the paper and giving constructive comments, which improved the manuscript substantially. The authors also acknowledge that this work is partially supported by the Universiti Putra Malaysia. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 3 November 216 Accepted: 31 March 217 References 1. Mainardi, F: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models 21) 2. Srivastava, HM: A contour integral involving Foxs H-function. Indian J. Math. 14, ) 3. Agarwal, P, Chand, M, Purohit, SD: A note on generating functions involving the generalied Gauss hypergeometric function. Nat. Acad. Sci. Lett. 375), ) 4. Agarwal, P, Koul, CL: On generating functions. J. Rajasthan Acad. Phys. Sci. 23), ) 5. Oergin, E: Some properties of hypergeometric functions. Ph.D. thesis, Eastern Mediterranean University, North Cyprus 211) 6. Srivastava, HM: Certain generating functions of several variables. Z. Angew. Math. Mech. 57, ) 7. Lee, DM, Rathie, AK, Parmar, R, Kim, YS: Generaliation of extended beta function, hypergeometric function and confluent hypergeometric functions. Honam Math. J. 33, ) 8. Chaudhry, MA, Qadir, A, Rafique, M, Zubair, SM: Extension of Euler s beta function. J. Comput. Appl. Math. 78, ) 9. He, JH: A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys. 53, ) 1. Liu, FJ, Li, ZB, Zhang, S, Liu, HY: He s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy. Therm. Sci. 19, ) 11. Yang, XJ, Baleanu, D, Srivastava, HM: Local Fractional Integral Transforms and Their Applications. Academic Press, Amsterdam 216) 12. Kiyma, IO, Cetinkaya, A, Agarwal, P: An extension of Caputo fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 96), ) 13. Daftardar-Gejji, V, Bhalekar, S: Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl. 345, ) 14. Jain, S, Choi, J, Agarwal, P: Generating functions for the generalied Appell function. Int. J. Math. Anal. 11), ) 15. Anderson, GD, Barnard, RW, Richards, KC, Vamanamurthy, MK, Vuorinen, M: Inequalities for ero-balanced hypergeometric functions. Trans. Am. Math. Soc. 3475), ) 16. Barnard, R, Richards, K: A note on the hypergeometric mean value. Comput. Methods Funct. Theory 11), ) 17. Karp, D, Sitnik, SM: Inequalities and monotonicity of ratios for generalied hypergeometric function. J. Approx. Theory 161, ) 18. Chand, M, Agarwal, P, Choi, J: Note on generating relations associated with the generalied Gauss hypergeometric function.appl.math.sci.135), ) 19. Li,H:LipschitspacesandQ K type spaces. Sci. China Math. 53, ) 2. Saxena, RK, Kalla, SL, Saxena, R: Multivariable analogue of generalied Mittag-Leffler function. Integral Transforms Spec. Funct. 227), ) 21. Chand, M, Guariglia, E: Note on Euler type integrals. Int. Bull. Math. Res. 22), ) 22. Djrbashian, MM, Zakaryan, VS: Classes and Boundary Properties of Functions Meromorphic in the Disc. Nauka, Moscow 1993) 23. Frederico, G, Torres, D: Fractional Noether s theorem in the Ries Caputo sense. Appl. Math. Comput. 217, ) 24. Agrawal, OP: Fractional variational calculus in terms of Ries fractional derivatives. J. Phys. A 424), )