New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials

Abstract

In this paper, by using q-Volkenborn integral [10], the first author [25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach to the complete sums of products of twisted (h, q)-Bernoulli polynomials and numbers. p-adic q-Volkenborn integral is used to evaluate summations of the following form:

A New Generating Function of (q‐) Bernstein‐Type Polynomials and Their Interpolation Function

“…So, the coefficients in all the usual cenral-difference formulae for interpolation, numerical differentiation and integration, and differences in terms of derivatives can be expressed in terms of these polynomials cf. ( [1], [10], [11], [20]). These polynomials are related to the many branches of Mathematics.…”

Multiple Interpolation Functions of Higher Order (h,q)‐Bernoulli Numbers

“…Simsek et al [14] defined twisted (/z, ig)-Bemoulli polynomials and numbers of higher-order as follows \t + hlogq \< 2n, cf. [14].…”

Bernoulli polynomials and Pascal matrices in the context of Clifford analysis

“…Consequently, bidimensional Bernoulli polynomials B (j) n (x, y), j ≥ 2 are therefore obtained by means of the generating function te xt+yt j e t −1 . Other recently obtained generalizations can be found in [14,15]. In this article we are following similar ideas combined with methods of generalized power series representations used in the theory of hypercomplex holomorphic (monogenic) functions, which are generalized complex holomorphic functions in the context of Clifford Analysis.…”