duce (p, q)âextended Bessel function Jν,p,q, the (p, q)âextended modified. Bessel function Iν,p,q of the first kind of order ν by making use two addi-.
(p, q)–extended Bessel and modified Bessel functions of the first kind Dragana Jankov Maˇsirevi´c, Rakesh K. Parmar and Tibor K. Pog´any
Abstract. Inspired by certain recent extensions of the Euler’s beta, Gauß hypergeometric and confluent hypergeometric functions [1], we introduce (p, q)–extended Bessel function Jν,p,q , the (p, q)–extended modified Bessel function Iν,p,q of the first kind of order ν by making use two additional parameters in the integrand, as well as the (p, q)–extended Struve Hν,p,q and the modified Struve Lν,p,q functions. Systematic investigation of its properties, among others integral representations, bounding inequalites Mellin transforms (for all newly defined Bessel and Struve functions), complete monotonicity, Tur´ an type inequality, associated non-homogeneous differential-difference equations (exclusively for extended Bessel functions) are presented. Brief presentation of another members of Bessel functions family: spherical ultrashperical, Delerue hyper–Bessel and their modified counterparts and the Wright generalized Bessel function with links to their (p, q)-extensions are proposed. Mathematics Subject Classification (2010). Primary 33B15,33C10,39B62; Secondary 26A48, 33E20. Keywords. (p, q)-extended Beta function; (p, q)-extended Bessel and modified Bessel functions of first kind; (p, q)-extended Struve and modified Struve functions; Integral representation; Bounding inequalities; Complete monotonicity; Differential–difference equation; Tur´ an type inequality.
1. Introduction and preliminaries Recently various extensions of the Euler function of the first kind (or Beta R1 function in other words) B(r, s) = 0 tr−1 (1 − t)s−1 dt, min{
