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 qbeta 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.

Rate of Approximation for Modified Lupaş-Jain-Beta Operators

“…As we know, in order to approximate Lebesgue integrable functions, the most important modifications are Kantorovich and Durrmeyer integral operators. Motivated by the above mentioned Durrmeyer type generalizations of various operators and also from [11][12][13][14][15][16][17][18][19][20][21][22][23], in this paper, Durrmeyer-type modification of generalized Lupaş-Jain operators (5) by taking weights of some beta basis function is defined as follows:…”

New Extension of Beta Function and Its Applications

“…In many areas of applied mathematics, different types of special functions have become necessary tool for the scientists and engineers. During the recent decades or so, numerous interesting and useful extensions of the different special functions (the Gamma and beta functions, the Gauss hypergeometric function, and so on) have been introduced by different authors [1][2][3][4][5][6].…”

Combinatorial sums and binomial identities associated with the Beta-type polynomials

“…These polynomials are used in combinatorics, in probability, in number theory, in mathematical analysis, and in algebra (cf. [22], [24], [21], [3]); thus, they have variety of important applications.…”