On one generalization of Bessel function

Abstract

In this paper the generalized Bessel function $J_{\mu ,\omega } ( x )$ is introduced. The function $J_{\mu ,\omega } ( x )$ is given as one solution of the following differential equation: $$ x^2{y}''+x{y}'+\left( {x-\mu ^2} \right)\left( {x+\omega ^2} \right)y=0, \quad \mu , \omega \notin \mathbb Z. $$ The representation of the $J_{\mu ,\omega } ( x )$ by the power series is given. The theorem on integral representations of the function $J_{\mu ,\omega } ( x )$ is established. The main properties of the function $J_{\mu ,\omega } ( x )$ are studied. The integral transforms of Bessel type with the function $J_{\mu ,\omega } ( x )$ is constructed. Formula of inversion of this transform is received.

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