Bessel equation and Bessel function on $\\mathbb{T}(q h)$

Abstract

This article is devoted to present nabla $(q h)$ -analogues of Bessel equation and Bessel function. In order to construct series solution of nabla $(q h)$ -Bessel equation we present nabla $(q h)$ -analysis regarding nabla generalized quantum binomial nabla $(q h)$ -analogues of Taylor’s formula Gauss’s binomial formula Taylor series analytic functions analytic exponential function with its fundamental properties analytic trigonometric and hyperbolic functions. We emphasize that nabla $(q h)$ -Bessel equation recovers classical $h-and q-discrete$ Bessel equations. In addition we establish nabla $(q h)4 -Bessel function which is expressed in terms of an absolutely convergent series in nabla generalized quantum binomials and intimately demonstrate its reductions. Finally we develop modified nabla $(q h)$ -Bessel equation modified nabla $(q h)$ -Bessel function and its relation with nabla $(q h)$ -Bessel function.