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...

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Main Authors: Rakesh K. Parmar, Ritu Agarwal, Naveen Kumar, S. D. Purohit
Format: Article
Language:English
Published: SpringerOpen 2021-08-01
Series:Advances in Difference Equations
Subjects:
Online Access:https://doi.org/10.1186/s13662-021-03536-0
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author Rakesh K. Parmar
Ritu Agarwal
Naveen Kumar
S. D. Purohit
author_facet Rakesh K. Parmar
Ritu Agarwal
Naveen Kumar
S. D. Purohit
author_sort Rakesh K. Parmar
collection DOAJ
description 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.
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spelling doaj.art-b6c3aa7a23424f6cacf18e0750f1f3882022-12-21T21:33:06ZengSpringerOpenAdvances in Difference Equations1687-18472021-08-012021111610.1186/s13662-021-03536-0Extended elliptic-type integrals with associated properties and Turán-type inequalitiesRakesh K. Parmar0Ritu Agarwal1Naveen Kumar2S. D. Purohit3Department of HEAS (Mathematics), University College of Engineering and TechnologyDepartment of Mathematics, Malaviya National Institute of TechnologyDepartment of Mathematics, Malaviya National Institute of TechnologyDepartment of HEAS (Mathematics), Rajasthan Technical UniversityAbstract 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.https://doi.org/10.1186/s13662-021-03536-0Turán-type inequalitiesElliptic integralsExtended beta functionExtended hypergeometric functionsMellin transformLaguerre polynomials
spellingShingle Rakesh K. Parmar
Ritu Agarwal
Naveen Kumar
S. D. Purohit
Extended elliptic-type integrals with associated properties and Turán-type inequalities
Advances in Difference Equations
Turán-type inequalities
Elliptic integrals
Extended beta function
Extended hypergeometric functions
Mellin transform
Laguerre polynomials
title Extended elliptic-type integrals with associated properties and Turán-type inequalities
title_full Extended elliptic-type integrals with associated properties and Turán-type inequalities
title_fullStr Extended elliptic-type integrals with associated properties and Turán-type inequalities
title_full_unstemmed Extended elliptic-type integrals with associated properties and Turán-type inequalities
title_short Extended elliptic-type integrals with associated properties and Turán-type inequalities
title_sort extended elliptic type integrals with associated properties and turan type inequalities
topic Turán-type inequalities
Elliptic integrals
Extended beta function
Extended hypergeometric functions
Mellin transform
Laguerre polynomials
url https://doi.org/10.1186/s13662-021-03536-0
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AT naveenkumar extendedelliptictypeintegralswithassociatedpropertiesandturantypeinequalities
AT sdpurohit extendedelliptictypeintegralswithassociatedpropertiesandturantypeinequalities