Turkish Journal of Mathematics




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.


Nabla generalized quantum binomial, nabla $(q, h)$-Taylor series, nabla $(q, h)$-analytic functions, nabla $(q, h)$-Bessel equation, nabla $(q, h)$-Bessel function

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