No Arabic abstract
Recently, various extensions and variants of Bessel functions of several kinds have been presented. Among them, the $(p,q)$-confluent hypergeometric function $Phi_{p,q}$ has been introduced and investigated. Here, we aim to introduce an extended $(p,q)$-Whittaker function by using the function $Phi_{p,q}$ and establish its various properties and associated formulas such as integral representations, some transformation formulas and differential formulas. Relevant connections of the results presented here With those involving relatively simple Whittaker functions are also pointed out.
In this study our aim to define the extended $(p,q)$-Mittag-Leffler(ML) function by using extension of beta functions and to obtain the integral representation of new function. We also take the Mellin transform of this new function in terms of Wright hypergeometric function. Extended fractional derivative of the classical Mittag-Leffler(ML) function leads the extended (p,q)-Mittag-Leffler(ML) function.
Our purpose in this present paper is to investigate generalized integration formulas containing the generalized $k$-Bessel function $W_{v,c}^{k}(z)$ to obtain the results in representation of Wright-type function. Also, we establish certain special cases of our main result.
In this present paper, we establish the log-convexity and Turan type inequalities of extended $(p,q)$-beta functions. Also, we present the log-convexity, the monotonicity and Turan type inequalities for extended $(p,q)$-confluent hypergeometric function by using the inequalities of extended $(p,q)$-beta functions.
In this paper we will establish some double-angle formulas related to the inverse function of $int_0^x dt/sqrt{1-t^6}$. This function appears in Ramanujans Notebooks and is regarded as a generalized version of the lemniscate function.
In this paper, we establish two new transformation formulas for ${}_{8}psi_{8}$ and ${}_8phi_7$ series by means of Slaters general transformation for bilateral series. As applications, some specific transformation formulas are presented among which include a general form of Weierstrass theta identity and new proofs of Baileys VWP ${}_6psi_6$ and Jacksons ${}_8phi_7$ summation formula.