This article aims to present (q; h)-analogue of exponential function which unifies, extends hand q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q,h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.

In this article, we focus our attention on (q,h)-Gauss?s binomial formula from which we discover the additive property of (q; h)-exponential functions. We state the (q,h)-analogue of Gauss?s binomial formula in terms of proper polynomials on T(q,h) which own essential properties similar to ordinary polynomials. We present (q,h)-Taylor series and analyze the conditions for its convergence. We introduce a new (q,h)-analytic exponential function which admits the additive property. As consequences, we study (q,h)-hyperbolic functions, (q,h)-trigonometric functions and their significant properties such as (q,h)-Pythagorean Theorem and double-angle formulas. Finally, we illustrate our results by a first order (q,h)-difference equation, (q,h)-analogues of dynamic diffusion equation and Burger?s equation. Introducing (q,h)-Hopf-Cole transformation, we obtain (q,h)-shock soliton solutions of Burger?s equation.

This article is devoted to present $(q,h)$-analogue of power function which satisfies additivity and derivative properties similar to the ordinary power function. In the light of nabla $(q,h)$-power function, we present $(q,h)$-analogue of binomial series and conclude that such power function is $(q,h)$-analytic. We prove the analyticity by showing that both the power function and its absolutely convergent Taylor series solve the same IVP. Finally, we present the reductions of $(q,h)$-binomial series to classical binomial series, Gauss' binomial and Newton's binomial formulas.

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.

In this paper, we consider the problem of quantization of classical St\"ackel systems and the problem of separability of related quantum Hamiltonians. First, using the concept of St\"ackel transform, all considered systems are expressed by flat coordinates of related Euclidean configuration space. Then, the so-called flat minimal quantization procedure is applied in order to construct an appropriate Hermitian operator in the respective Hilbert space. Finally, we distinguish a class of St\"ackel systems which remains separable after any of admissible flat minimal quantizations.Comment: 12 page

On a general time scale, polynomials, Taylors formula, and related subjects are described in terms of implicit and inefficient recursive relations. In this work, our primary goal is to construct proper polynomials, namely delta and nabla generalized quantum polynomials, on (q, h)-time scales explicitly. We show that generalized quantum polynomials play the same roles on (q, h)-time scales as ordinary polynomials play in $\mathbb{R}$ since they obey the additive properties and Leibnitz rules. Such polynomials which recover falling/rising and q-falling/q-rising factorials are constructed by the frame of forward and backward shifts. Additionally, we present delta- and nabla-Gauss binomial formulas which provide many applications.