Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 191–200 DOI: 10.18514/MMN.2018.1503 THE INCOMPLETE SRIVASTAVA’S TRIPLE HYPERGEOMETRIC FUNCTIONS  H A AND  H A JUNESANG CHOI AND RAKESH K. PARMAR Received 20 January, 2015 Abstract. Motivated mainly by certain interesting recent extensions of the generalized hyper- geometric function [15], the second Appell function [6] and Srivastava’s triple hypergeometric functions [9], we introduce here the family of incomplete Srivastava’s triple hypergeometric func- tions  H A and  H A . We then systematically investigate several properties of each of these incom- plete Srivastava’s triple hypergeometric functions including, for example, their various integral representations, transformation formula, reduction formula, derivative formula and recurrence relations. Various (known or new) special cases and consequences of the results presented here are also considered. 2010 Mathematics Subject Classification: 33B15; 33B20; 33C05; 33C15; 33C20; 33B99; 33C99; 60B99 Keywords: incomplete gamma function, incomplete Pochhammer symbol, incomplete gener- alized hypergeometric functions, incomplete second Appell function, Srivastava’s triple hyper- geometric functions, Laguerre polynomials, Bessel and modified Bessel functions, incomplete Srivastava’s triple hypergeometric functions 1. I NTRODUCTION, DEFINITIONS AND PRELIMINARIES The familiar incomplete Gamma functions .s;x/ and .s;x/ defined by .s;x/ WD Z x 0 t s1 e t dt  <.s/ > 0I x = 0  (1.1) and .s;x/ WD Z 1 x t s1 e t dt  x = 0I<.s/ > 0 when x D 0  ; (1.2) respectively, satisfy the following decomposition formula: .s;x/ C .s;x/ WD  .s/  <.s/ > 0  : (1.3) Each of these functions plays an important role in the study of the analytic solutions of a variety of problems in diverse areas of science and engineering (see, e.g.,[1, 4, 7, 16, 17, 23]). c  2018 Miskolc University Press