Appl. Math. Inf. Sci. 9, No. 4, 2161-2167 (2015) 2161 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090456 Certain Integral Transforms for the Incomplete Functions Junesang Choi 1 and Praveen Agarwal 2,∗ 1 Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea 2 Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India Received: 25 Nov. 2014, Revised: 25 Feb. 2015, Accepted: 27 Feb. 2015 Published online: 1 Jul. 2015 Abstract: Srivastava et al. [14] introduced the incomplete Pochhammer symbols that lead to a natural generalization and decomposition of a class of hypergeometric and other related functions to mainly investigate certain potentially useful closed-form representations of definite and semi-definite integrals of various special functions. Here, in this paper, we use the integral transforms like Beta transform, Laplace transform, Mellin transform, Whittaker transforms, K-transform and Hankel transform to investigate certain interesting and (potentially) useful integral transforms the incomplete hypergeometric type functions p γ q [z] and p Γ q [z]. Relevant connections of the various results presented here with those involving simpler and earlier ones are also pointed out. Keywords: Gamma function; Beta function; Incomplete gamma functions; Incomplete Pochhammer symbols; incomplete hypergeometric functions; Beta transform; Laplace transform; Mellin transform; Whittaker transforms; K-transform; Hankel transform 1 Introduction and Preliminaries The incomplete Gamma type functions like γ (s, x) and Γ (s, x), both of which are certain generalizations of the classical Gamma function Γ (z), given in (1) and (2), respectively, have been investigated by many authors. The incomplete Gamma functions have proved to be important for physicists and engineers as well as mathematicians. For more details, one may refer to the books [1, 2, 5, 20, 22] and the recent papers [3, 13, 14, 17, 18] and [19] on the subject. The familiar incomplete Gamma functions γ (s, x) and Γ (s, x) defined by γ (s, x) :=  x 0 t s−1 e −t dt (ℜ(s) > 0; x ≥ 0), (1) and Γ (s, x) :=  ∞ x t s−1 e −t dt (2) (x ≥ 0; ℜ(s) > 0 when x = 0), respectively, satisfy the following decomposition formula γ (s, x)+ Γ (s, x)= Γ (s) (ℜ(s) > 0), where Γ (s) is the well-known Euler’s Gamma function defined by Γ (s) :=  ∞ 0 t s−1 e −t dt (ℜ(s) > 0). (3) We also recall the Pochhammer symbol (λ ) n defined (for λ ∈ C) by (λ ) n : =  1 (n = 0) λ (λ + 1) ··· (λ + n − 1) (n ∈ N := {1, 2, ...}) = Γ (λ + n) Γ (λ ) (λ ∈ C \ Z − 0 ), (4) where Z − 0 denotes the set of nonpositive integers (see, e.g., [15, p. 2 and p. 5]). The theory of the incomplete Gamma functions, as a part of the theory of confluent hypergeometric functions, has received its first systematic exposition by Tricomi [21] in the early 1950s. Musallam and Kalla [8, 9] considered a more general incomplete gamma function involving the Gauss hypergeometric function and established a number of analytic properties including recurrence relations, asymptotic expansions and computation for special values of the parameters. Very recently, Srivastava et al.[14] introduced and studied some fundamental properties and characteristics of a family of two potentially useful and generalized incomplete hypergeometric functions defined as follows: ∗ Corresponding author e-mail: praveen.agarwal@anandice.ac.in, goyal.praveen2011@gmail.com c  2015 NSP Natural Sciences Publishing Cor.