Research Article Further Results on the (p, k) - Analogue of Hypergeometric Functions Associated with Fractional Calculus Operators Muajebah Hidan , 1 Salah Mahmoud Boulaaras , 2,3 Bahri-Belkacem Cherif , 2,4 and Mohamed Abdalla 1,5 1 Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia 2 Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Buraydah, Saudi Arabia 3 Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, 31000 Oran, Algeria 4 Preparatory Institute for Engineering Studies in Sfax, Sfax, Tunisia 5 Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt Correspondence should be addressed to Bahri-Belkacem Cherif; bahi1968@yahoo.com Received 24 February 2021; Revised 8 March 2021; Accepted 15 March 2021; Published 31 March 2021 Academic Editor: A. M. Nagy Copyright © 2021 Muajebah Hidan et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In a previous article, first and last researchers introduced an extension of the hypergeometric functions which is called “(p, k)-extended hypergeometric functions.” Motivated by this work, here, we derive several novel properties for these functions, including integral representations, derivative formula, k-Beta transform, Laplace and inverse Laplace transforms, and operators of fractional calculus. Relevant connections of some of the discussed results here with those presented in earlier references are outlined. 1.Introduction Recently, various extensions of the hypergeometric func- tions have been presented and investigated (see, for example, [1–7]). In particular, Diaz and Pariguan [8] introduced inter- esting generalizations of the gamma, beta, Pochhammer, and hypergeometric functions as follows. Definition 1. For k ∈ R + , the k-gamma function Γ k (w) is defined by Γ k (w)�  ∞ 0 z w− 1 e − w k /k ( ) dz, w ∈ C\kZ − . (1) We note that Γ k (w) ⟶Γ(w), for k ⟶ 1, where Γ(w) is the classical Euler’s gamma function and (w) m,k is the k-Pochhammer symbol given by (w) m,k � Γ k (w + mk) Γ k (w) � w(w + k) ... (w +(m − 1)k), m ∈ N,w ∈ C, 1, m � 0,k ∈ R + ,w ∈ C\ 0 {}. ⎧ ⎪ ⎨ ⎪ ⎩ (2) e relation between the Γ k (w) and the usual gamma function Γ(w) (see, e.g., [9]) follows easily that Γ k (w)� k (w/k)− 1 Γ w k , or Γ(τ)� k 1− τ Γ k (kτ). (3) Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 5535962, 10 pages https://doi.org/10.1155/2021/5535962