Author:
Parmar Rakesh K.,Agarwal Ritu,Kumar Naveen,Purohit S. D.
Abstract
AbstractOur 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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Algebra and Number Theory,Analysis
Cited by
2 articles.
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