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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Format: | Article |
Language: | English |
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SpringerOpen
2021-08-01
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Series: | Advances in Difference Equations |
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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. |
first_indexed | 2024-12-17T20:48:41Z |
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id | doaj.art-b6c3aa7a23424f6cacf18e0750f1f388 |
institution | Directory Open Access Journal |
issn | 1687-1847 |
language | English |
last_indexed | 2024-12-17T20:48:41Z |
publishDate | 2021-08-01 |
publisher | SpringerOpen |
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series | Advances in Difference Equations |
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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