Some families of generalized complete and incomplete elliptic-type integrals

Analogous to the recent generalizations of the familiar beta and hypergeometric functions by Lin et al. [S.-D. Lin, H. M. Srivastava, J.-C. Yao, Appl. Math. Inform. Sci., 9 (2015), 1731–1738], the authors introduce and investigate some general families of the elliptic-type integrals for which the usual properties and representations are naturally and simply extended. The object of the present paper is to study these generalizations and their relationships with generalized hypergeometric functions of one, two and three variables. Moreover, the authors establish the Mellin transform formulas and various derivative and integral properties and obtain several relations for special cases in terms of well-known higher transcendental functions and some infinite series representations containing the Meijer G-function, the Whittaker function and the complementary error functions, as well as the Laguerre polynomials and the products thereof. A number of (known or new) special cases and consequences of the main results presented here are also considered.