Q-Beta Polynomials and their Applications

Abstract

The aim of this paper is to construct generating functions for q-beta polynomials. By using these generating functions, we define the q -beta polynomials and also derive some fundamental properties of these polynomials. We give some functional equations and partial differential equations (PDEs) related to these generating functions. By using these equations, we find some identities related to these polynomials, binomial coefficients, the gamma function and the beta function. We obtain a relation between the q- beta polynomials and the q-Bernstein basis functions. We give relations between the q-Beta polynomials, the Bernoulli polynomials, the Euler polynomials and the Stirling numbers. We also give a probability density function associated with the beta polynomials. By applying the Mellin transform, the Fourier transform and the Laplace transform to the generating functions, we obtain not only interpolation function, but also some series representations for the q-Beta polynomials. Furthermore, by using the p-adic q-Volkenborn integral, we give relations between the q-beta polynomials, the q-Euler numbers and the Carlitz's q-Bernoulli numbers.