Certain Generating Functions Involving Generalized Mittag-Leffler Function

International Journal of Mathematical Analysis Vol. 12, 2018, no. 6, 269-276 HIKARI Ltd, www.m-hiari.com https://doi.org/10.12988/ijma.2018.8431 Certain Generating Functions Involving Generalized Mittag-Leffler Function Mehar Chand and Reha Rani Department of Applied Sciences Guru Kashi University Bathinda-1513002, India Copyright c 2018 Mehar Chand and Reha Rani. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract The objective of the present paper is to establish some new generating relations involving the generalized Mittag-Leffler function. Our main results can yield a large number of new interesting generating relations of related special functions. Mathematics Subject Classification: 26A33, 44A20, 33E12, 44A10 Keywords: Generating functions, the generalized Mittag-Leffler function, -Pochhemmer symbol, Taylor s theorem, Hypergeometric function 1 Introduction and Preliminaries We begin by recalling some basic results. Here and in the following, let C, R, R +, N and Z 0 be the sets of complex numbers, real numbers, positive real numbers, positive integers and non-positive integers, respectively. Diaz and Pariguan [7] introduced the -Pochhemmer symbol as follows: (γ n, = Γ (γ + n (n N; R + ; γ C \ {0}, Γ (γ γ(γ +...(γ + (n 1 (n N; γ C (1

270 Mehar Chand and Reha Rani where Γ is the -gamma function which has the following relation with the classical Euler s gamma function (see, e.g., [17]. For γ C, R + ( Γ (γ = γ γ 1 Γ. (2 For = 1, Eq. e.g.,[17]. (1 reduces to the classical Pochhammer symbol (see, (γ n = Γ(γ + n Γ(γ = { 1 (n = 0; γ C \ {0}, γ(γ + 1...(γ + (n 1 (n N; γ C (3 Let α, β, γ C; R(α > 0, R(β > 0, R(γ > 0 and q R +, then the generalized Mittag-Leffler function, denoted by E γ,q α,β (z, is defined as (Shula and Prajapati [14] E γ,q α,β (z = (γ nq z n Γ(nα + β n!. (4 For q = 1 Eq. (4 gives Mittag-Leffler function, defined as (Dorrego and Cerutti [8] E γ,1 α,β (z = (γ n z n Γ(nα + β n!. (5 The concept of the Hadamard product (or the convolution of two analytic functions is very useful in our present study. Let and f(z := g(z := a n z n ( z < R f (6 b n z n ( z < R g (7 be two power series whose radii of convergence are denoted by R f and R g, respectively. Then, their Hadamard product is the power series defined by (f g(z := a n b n z n = (g f(z ( z < R (8 ( R = lim a n b n n a n+1 b n+1 = lim n a n a n+1 (. lim n b n b n+1 = R f.r g, (9

Certain generating functions involving generalized Mittag-Leffler function 271 therefore, in general, we have R R f.r g [9, 12]. For various investigations involving the Hadamard product (or the convolution, the interested reader may refer to several recent papers on the subject (see, for example, [15, 16] and the references cited therein. Also we require the Fox-Wright function p Ψ q (z (p, q N 0 with p numerator and q denominator parameters defined for a 1,..., a p C and b 1,..., b q C \ Z 0 by (For details see [10, 11, 13, 18] pψ q [ (a1, α 1,..., (a p, α p ; (b 1, β 1,..., (b q, β q ; ] z = Γ(a 1 + α 1 n...γ(a p + α p n z n Γ(b 1 + β 1 n...γ(b q + β q n n!, (10 where the coefficients α 1,..., α p, β 1,..., β q R + are such that 1 + q β j j=1 p α i 0 (11 For α i = β j = 1 (i = 1,..., p; j = 1,..., q, Eq. (10 reduces immediately to the generalized hypergeometric function p F q (p, q N 0 (see [18]: [ ] a1,..., a p ; pf q b 1,..., b q ; z = Γ(b [ ] 1...Γ(b q (a1, 1,..., (a Γ(a 1...Γ(a q p Ψ p, 1; q (b 1, 1,..., (b q, 1; z (12 The generalized hypergeometric series p F q (. is defined as: [ ] α1,..., α p ; pf q β 1,..., β q ; z = i=1 (α 1 n...(α p n z n (β 1 n...(β q n n! = p F q (α 1,..., α p ; β 1,..., β q ; z, (13 Generating functions are widely used in the study of various daily life phenomenon. In this regard, many researchers has developed a remarably large no. of interesting generating functions involving a variety of special functions. For more details one can refer the extensive wor of Srivastava and Manocha [19], Agarwal et. al. [1, 2, 3], see also [4, 5, 6, 18, 20, 21, 22, 23]. Motivated from the above wor, here we aim to establish some new generating relations involving generalized Mittag-Leffler function by applying Taylor s theorem. The main results presented here are general enough to yield a large number of new, interesting and useful generating functions. Consider a function g(s = s λ nξ (λ, ξ C, n N. Then it is easy to find that the -th derivative of g(s can be expressed in terms of the Gamma functions as follows: g (s = ( 1 λ nξ Γ(λ + nξ + s ( N 0. (14 Γ(λ + nξ

272 Mehar Chand and Reha Rani 2 Main Results In this section, we establish some generating relations involving the generalized Mittag-Leffler function presented in the following theorems. Theorem 2.1. Letα, β, γ C; R(α > 0, R(β > 0, R(γ > 0 and q R +, t < 1; then the following generating functions hold: (1 + t σ E γ,q α,β (z/(1 + t = ( 1 (σ E γ,q α,β (z 1F 1 (σ + ; σ; z t!. (15 Proof: For convenience, we replace 1 + t by s and denote left hand side of the result (15 by g(s, then by using (4, we have g(s = s σ E γ,q α,β (z/s = (γ nq z n Γ(nα + β n! s σ n (16 Differentiating times both sides of (16 with respect to s, we have g (s = ( 1 s σ (γ nq Γ(σ + n + ( z n 1 Γ(nα + β Γ(σ + n s n! After a little simplification, Eq. (17 reduces to (17 g (s = ( 1 s σ (σ (γ nq (σ + ( n z n 1 Γ(nα + β (σ n s n! And interpreting the above equation (18 with the help of (8, we have (18 g (s = ( 1 s σ (σ E γ,q α,β (z/s 1F 1 (σ + ; σ; z/s. (19 Expanding g(s + t as the Taylor series, we have g(s + t = Using Eqs. (16 and (19, we get t! g( (s. (20 ( t s σ! (21 If we choose s = 1 for Eq. (21, we arrive at the required result (15. (s + t σ E γ,q α,β (z/(s + t = (σ E γ,q α,β (z/s 1F 1 (σ + ; σ; z/s.

Certain generating functions involving generalized Mittag-Leffler function 273 Theorem 2.2. Letα, β, γ C; R(α > 0, R(β > 0, R(γ > 0 and q R +, t < 1; then the following generating functions hold: ( γ + 1 ( z E γ,q α,β (zt = (1 t γ E γ,q α,β. (22 (1 t q Proof: For convenience, let we denote left hand side of the result (22 by J, then by using (4, we have J = ( { γ + 1 (γ + nq Γ(nα + β } z n t (23 n! By interchanging the order of summations and using the following nown identity [17] ( γ = Γ(γ + 1!Γ(γ + 1 And after little simplification, we have J = (γ nq Γ(nα + β ( N 0, γ C (24 { ( } γ + nq + 1 t z n n! (25 And using the generalized binomial expansion, we find that the inner sum in Eq. (25, gives ( γ + nq + 1 t = (1 t (γ+nq ( t < 1. (26 And interpreting the above Eq. (25 with the help of (26, we reach at the required result (22 3 Special Cases If we set q = 1, the above results (15 and (22 reduce to the following form: Corollary 3.1. Letα, β, γ C; R(α > 0, R(β > 0, R(γ > 0, t < 1; then the following generating functions hold: (1 + t σ E γ α,β (z/(1 + t = ( 1 (σ E γ α,β (z 1F 1 (σ + ; σ; z t!. (27

274 Mehar Chand and Reha Rani Corollary 3.2. Letα, β, γ C; R(α > 0, R(β > 0, R(γ > 0, t < 1; then the following generating functions hold: ( γ + 1 References ( z E γ α,β (zt = (1 t γ E γ α,β. (28 (1 t [1] P. Agarwal and C.L. Koul, On generating functions, J. Rajasthan Acad. Phy. Sci., 2 (2003, no. 3, 173-180. [2] P. Agarwal, M. Chand and S. D. Purohit, A note on generating functions involving the generalized Gauss hypergeometric function, National Academy Science Letters, 37 (2014, no. 5, 457-459. https://doi.org/10.1007/s40009-014-0250-7 [3] P. Agarwal, M. Chand and G. Singh, On new sequence of functions involving the product of Fp α,β (., Communications in Numerical Analysis, 2015 (2015, no. 2, 139-148. https://doi.org/10.5899/2015/cna-00248 [4] M. P. Chen and H. M. Srivastava, Orthogonality relations and generating functions for Jacobi polynomials and related hypergeometric functions, Appl. Math. Comput., 68 (1995, 153-188. https://doi.org/10.1016/0096-3003(9400092-i [5] J. Choi and P. Agarwal, Certain Class of Generating Functions for the Incomplete Hypergeometric functions, Abstract and Applied Analysis, 2014 (2014, Article ID 714560 1-5. https://doi.org/10.1155/2014/714560 [6] M. Chand, P. Agarwal and J. Choi, Note on Generating Relations Associated with the Generalized Gauss Hypergeometric Function, Applied Mathematical Sciences, 10 (2016, no. 35, 1747-1754. https://doi.org/10.12988/ams.2016.6386 [7] R. Diaz and E. Pariguan, On hypergeometric functions and - Pochhammer symbol, Divulgaciones Mathematicas, 15 (2007, no. 2, 179-192. [8] G. A. Dorrego and R. A. Cerutti, The -Mittag-Leffler function, Int. J. Contemp. Math. Sci., 7 (2012, no. 15, 705-716. [9] V. Kiryaova, On two Saigos fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal., 9 (2006, 159 176.

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276 Mehar Chand and Reha Rani [21] H. M. Srivastava, Certain generating functions of several variables, Z. Angew. Math. Mech., 57 (1977, 339-340. https://doi.org/10.1002/zamm.19770570611 [22] H. M. Srivastava, Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions, Springer Plus, 2 (2013, 1-14. https://doi.org/10.1186/2193-1801-2-67 [23] H. M. Srivastava, A new family of the λ- generalized Hurwitz-Lerch Zeta functions with applications, Appl. Math. Inform. Sci., 8 (2014, 1485-1500. https://doi.org/10.12785/amis/080402 Received: May 7, 2018; Published: May 29, 2018