Commun. Korean Math. Soc. 32 (2017), No. 1, pp. 29–38 https://doi.org/10.4134/CKMS.c150247 pISSN: 1225-1763 / eISSN: 2234-3024 CERTAIN INTEGRALS ASSOCIATED WITH GENERALIZED MITTAG-LEFFLER FUNCTION Praveen Agarwal, Junesang Choi, Shilpi Jain, and Mohammad Mehdi Rashidi Abstract. The main objective of this paper is to establish certain uni- fied integral formula involving the product of the generalized Mittag- Leffler type function E (γ j ),(l j ) (ρ j ),λ [z 1 ,...,zr ] and the Srivastava’s polynomi- als S m n [x]. We also show how the main result here is general by demon- strating some interesting special cases. 1. Introduction and preliminaries Throughout this paper let C, R + 0 , N and Z − 0 denote the sets of complex numbers, non-negative real numbers, positive and non-positive integers, re- spectively, and N 0 := N ∪{0}. Since, in 1903, the Swedish mathematician Gosta Mittag-Leffler [12] introduced the function E α (z ) defined by (1.1) E α (z )= ∞  n=0 z n Γ(αn + 1) ( z ∈ C; α ∈ R + 0 ) , Γ(·) being the familiar Gamma function (see, e.g., [21, Section 1.1]), it has been actively investigated by many authors who have extended (or generalized) by adding parameters and variables to the previous extension and showed their importance by realizing a variety of applications in such research subjects as (for example) in physics, chemistry, biology, engineering and applied sciences (see, e.g., [7, 10, 11, 17]). Mittag-Leffler function also occurs as the solution of fractional order differential equation or fractional order integral equations. It is easy to see that E 0 (z ) in (1.1) reduces to a simple geometric series. For 0 <α< 1, the E α (z ) in (1.1) interpolates between the exponential function e z and the geometric function 1/(1 − z ). Here a brief historical account of the extensions of the Mittag-Leffler function E α (z ) in (1.1). In 1905, Wiman [25] introduced to investigate the following Received December 26, 2015. 2010 Mathematics Subject Classification. Primary 65A05, 33C45; Secondary 44A15. Key words and phrases. Gamma function, Srivastava polynomials, Hermite polynomi- als, Laguerre polynomials, generalized Mittag-Leffler type functions, Lauricella’s multiple hypergeometric series, Psi (or Digamma) function ψ(z). c 2017 Korean Mathematical Society 29