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Register a new userCertain unified integration formulas associated with generalized k-Bessel function
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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.
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We aim to introduce the generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $ and to present some formulas of the Riemann-Liouville fractional integration and differentiation operators. Further, we also derive certain integral formulas involving the newly defined generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $. We prove that such integrals are expressed in terms of the Fox-Wright function $_{p}Psi_{q}(z)$. The results presented here are of general in nature and easily reducible to new and known results.
Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz emph{et al.} [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. {bf 38} (2015), no.~3, 1255--1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results.
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 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.
With respect to generalized trigonometric functions, since the discovery of double-angle formula for a special case by Edmunds, Gurka and Lang in 2012, no double-angle formulas have been found. In this paper, we will establish new double-angle formulas of generalized trigonometric functions in two special cases.
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