Abstract
Our aim is to study and investigate the family of (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)-extended Gauss’ hypergeometric function and (p, q)-extended Appell’s double hypergeometric function F_{1}. Turán-type inequalities including log-convexity properties are proved for these (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)-extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with (p, q)-extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce (p, q)-extension of the Epstein–Hubbell (E-H) elliptic-type integral.
Highlights
Our aim is to study and investigate the family of (p, q)-extended elliptic-type integrals for which the usual properties and representations of various known results of the elliptic integrals are extended in a simple manner
Elliptic-type integrals such as Legendre elliptic integrals, generalized complete elliptic integrals of the first and second kind, and symmetric elliptic integrals [7] and several definite integrals of such families are known to play a prominent role in special functions in terms of their modulus or complementary modulus in the theory of conformal mappings [2], studies of crystallographic minimal surfaces, radiation physics problems [3], nuclear technology, fracture mechanics studies of elliptical crack problems, the study of electromagnetic or acoustic waves being scattered by an elliptic disk [15], astronomy, geometry, physics, and engineering mechanics [4]
The goal of this paper is to introduce and investigate the family of (p, q)-extended elliptic-type integrals and elliptic-type integrals, which are analogous on the basis of definition (1.7) of the (p, q)-extended beta function B(δ, σ ; p, q) so that many of the known properties of the elliptic-type integrals carry over naturally
Summary
Our aim is to study and investigate the family of (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)-extended Gauss’ hypergeometric function and (p, q)-extended Appell’s double hypergeometric function F1. 5, we provide certain connections with the (p, q)-extended beta function and Meijer’s G-function of two variables as new representations for the parameter(special) values and differential and integral properties of the (p, q)-extended(complete) elliptic-type integrals. Proof By using the definition of the classical Hölder–Rogers inequality for integrals in the integral representation (2.4), we have
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