In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on $(q;h)$-time scales. Our proposed approach encompasses both difference and quantum problems, making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator $\alpha$ that combines both jump operators in a convex manner. This allows us to generate a time scale $\alpha$, which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale $\alpha$, which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on $\alpha$-time scales using the $\alpha$-operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the $\alpha$-version of the Taylor formula. Finally, we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus.

This article is dedicated to present various concepts on α -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the α -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the α -Gauss binomial formula. Furthermore, we define the α -gamma function and prove α -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the α -gamma function is the unique logarithmically convex solution of f ( s + 1 ) = ϕ ( s ) f ( s ) , f ( 1 ) = 1 , where ϕ ( s ) refers to the α -number. In addition, we present Euler’s infinite product form and asymptotic behavior of α -gamma function. As an application, we propose α -analogue of the cylindrical diffusion equation, from which α -Bessel and modified α -Bessel equations are derived. We explore the solutions of the α -cylindrical diffusion equation using the separation of variables technique, revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally, we illustrate the graphs of the α -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.

This article introduces a novel approach for discrete Bessel equations and Bessel functions of the first and second kinds on $\alpha$-time scale. This approach not only addresses existing gaps in discrete, $q$-discrete and $(q,h)$-discrete Bessel equations and functions of the second kind but also provides their unification and extension. We introduce $\alpha$-Bessel equation of order $[\nu]\alpha$, construct $\alpha$-Bessel function of order $[\nu]\alpha$ of the first kind, $J_{\nu}^{\alpha}$, and analyze second linearly independent solutions in two distinct cases. We introduce $\alpha$-Bessel function of order $[\nu]\alpha$ of the second kind, namely $\alpha$-Weber's Bessel function, $Y{\nu}^{\alpha}$, as a special linear combination of $J_{\nu}^{\alpha}$ and $\bar{Y}{\nu}^{\alpha}$, where $\bar{Y}{\nu}^{\alpha}$ is $\alpha$-Neumann's Bessel function. The $\alpha$-Bessel function of the second kind recovers the various Bessel functions of the second kind in proper limits. Additionally, we present modified $\alpha$-Bessel equation and functions of the first and second kinds and state a relationship between first kinds of $\alpha$-Bessel and modified $\alpha$-Bessel functions. As an application, we introduce the $\alpha$-analogue of the Laplace equation in cylindrical coordinates and derive its solutions in terms of $\alpha$-analogues of trigonometric, hyperbolic, Bessel, and modified Bessel functions. Finally, graphs of some $\alpha$-Bessel functions are presented and reductions to several discrete Bessel equations and functions are outlined.

This manuscript investigates hypergeometric functions on a general discrete time scale $\mathbb{T}_\alpha$. To this end, we introduce the $\alpha$-beta function, establish its fundamental properties, and explore its connection with the $\alpha$-gamma function. In order to achieve our main objective, we formulate the $\alpha$-analogue of the hypergeometric equation and construct its solutions, referred as the $\alpha$-hypergeometric functions. In parallel with the classical case, we illustrate that the $\alpha$-hypergeometric function encompasses many $\alpha$-elementary functions. Furthermore, we derive several key properties of the $\alpha$-hypergeometric function, including the $\alpha$-analogues of the Gauss summation formula, Pfaff transformation, and Euler’s linear identity, which reduce to their classical and $q$-analogous forms.