Generalized Polynomials and Their Unification and Extension to Discrete Calculus

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.