Advances in Difference Equations | |
Extended elliptic-type integrals with associated properties and Turán-type inequalities | |
S. D. Purohit1  Rakesh K. Parmar2  Naveen Kumar3  Ritu Agarwal3  | |
[1] Department of HEAS (Mathematics), Rajasthan Technical University;Department of HEAS (Mathematics), University College of Engineering and Technology;Department of Mathematics, Malaviya National Institute of Technology; | |
关键词: Turán-type inequalities; Elliptic integrals; Extended beta function; Extended hypergeometric functions; Mellin transform; Laguerre polynomials; | |
DOI : 10.1186/s13662-021-03536-0 | |
来源: DOAJ |
【 摘 要 】
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.
【 授权许可】
Unknown