Extended elliptic-type integrals with associated properties and Turán-type inequalities

Abstract

Our aim is to study and investigate the family of ( p , q ) $(p, q)$ -extended ( incomplete and complete ) elliptic-type integrals for which the usual properties and representations of various known results of the ( classical ) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with ( p , q ) $(p, q)$ -extended Gauss’ hypergeometric function and ( p , q ) $(p, q)$ -extended Appell’s double hypergeometric function F 1 $F_{1}$ . Turán-type inequalities including log-convexity properties are proved for these ( p , q ) $(p, q)$ -extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these ( p , q ) $(p, q)$ -extended elliptic-type integrals and Meijer G -function of two variables. Moreover, we obtain several connections with ( p , q ) $(p, q)$ -extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce ( p , q ) $(p, q)$ -extension of the Epstein–Hubbell (E-H) elliptic-type integral.