Bessel equation and Bessel function on 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)-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.