Extended elliptic-type integrals with associated properties and Turán-type inequalities

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. 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.

1t 2 dt, (1.1) Obviously, their special cases are (1 - and well-known (canonical) Legendre incomplete elliptic integrals F(ψ, k) and E(ψ, k) and complete elliptic integrals K(k) and E(k) of the first and second kind (in terms of modulus |k| and amplitude ψ) [6]: In recent years, extensions of a number of well-known special functions have been investigated and studied the (p, q)-variant, and in turn, when p = q the p-variant together with the set of related higher transcendental hypergeometric type special functions (see, for details, [8-10, 17, 18, 20, 22-24]). In what follows we shall use the following recently defined (p, q)-extensions of the classical beta function B(x, y) and classical Gauss's hypergeometric function F(λ, μ; ν; Z) [11, p.
The goal of this paper is to introduce and investigate the family of (p, q)-extended (incomplete) elliptic-type integrals and (complete) 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. In Sect. 2, we introduce a family of (p, q)-extended elliptic-type integrals. The (p, q)-extension pro-

Log-convexity properties and Turán-type inequalities
In this section, we establish the Turán-type inequalities based upon log-convexity properties for the H p,q (k, γ ), K p,q (k), and E p,q (k) in (2.4), (2.5), and (2.6).

Theorem 3.1
The following assertions are true for (p) > 0, (q) > 0: . Moreover, for the same parametric range, the following Turán inequalities hold true: Proof By using the definition of the classical Hölder-Rogers inequality for integrals in the integral representation (2.4), we have This is equivalent to which proves the first assertion.
In a similar manner, by using (2.4) and using the Hölder-Rogers inequality, we get  (

Mellin transform formulas and Laguerre polynomial representations
The Mellin transforms of the function f (x, y) of two variables with respect to the indices r and s are given by [19] M f (x, y) (r, s) = where it is assumed that the integral (improper) in (4.1) exists.

Certain properties of (p, q)-extended elliptic integrals
In this section we obtain certain special values in terms of (p, q)-extended beta function B p,q (δ, σ ) and present the connections with G-function of two variables. We also present various derivative and integrals formulas for the (p, q)-extended elliptic-type integrals.

Special parametric values and connections with G-function
In this subsection, we first find the special values of K p,q (k), K p,q (k), E p,q (k), and E p,q (k) in terms of the (p, q)-extended beta function B p,q (δ, σ ). It suffices to consider the corresponding defining expressions in Sect. 2 in view of the definition of B p,q (δ, σ ) (1.7).

Differential and integral formulas
In this subsection, we present various differential and integral formulas for (p, q)-extended elliptic-type integrals. The proofs are omitted.

Theorem 5.2
The following derivative formulas hold true for K p,q (k) and E p,q (k):

Concluding remark and observations
In our present studies, we have introduced and extensively investigated the family of (incomplete and complete) (p, q)-extended elliptic-type integrals and presented connections with (p, q)-extended beta function, (p, q)-extended Gauss' hypergeometric function, and (p, q)-extended Appell's double hypergeometric function F 1 . Moreover, we obtained the connection with Meijer G-function of two variables. Turán-type inequalities were proved by using log-convexity property for these (p, q)-extended complete elliptic-type integrals. Further, we established various Mellin transform formulas and obtained certain infinite series representations containing Laguerre polynomials. We also obtained some relationship between these (p, q)-extended elliptic-type integrals and as special values and deduced numerous differentiation and integral formulas. In conclusion, we introduced (p, q)extension of the Epstein-Hubbell elliptic-type integral.
It is worth mentioning, as a main conclusion to observe, that Epstein and Hubbell [12] studied and investigated the following extensions of K(k) and E(k), which was encountered in a Legendre polynomial expansion method when applied to certain problems involving computation of the radiation field off-axis from a uniform circular disk radiating according to an arbitrary angular distribution law [14] (see also Weiss [37]): Note that, by comparing definitions (1.5), (1.6), and (6.1), we can deduce the following connections: E(k) k 2 := 2κ 2 1 + κ 2 .