On hypergeometric Bernoulli numbers and polynomials

Abstract

Abstract. In this note, we shall provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.

On $q$-hypergeometric Bernoulli polynomials and numbers

“…In this paper we have introduced two kinds of q-hypergeometric Bernoulli polynomials that generalized the hypergeometric Bernoulli polynomials introduced in [6], extensively studied in [10,14] and the reference therein. Classical results are obtained from the results of this paper by taking the limit as q tends to 1.…”

A new class of higher order hypergeometric Bernoulli polynomials associated with Hermite polynomials

“…For N, r ∈ N, the higher-order hypergeometric Bernoulli polynomials B (r) N,n (x) are defined by means of the generating function, (see [2], [7], [10])…”

Generalized hypergeometric Bernoulli numbers

“…In [10] a result of Kamano on a multiple binomial sum is generalized, higher order hypergeometric Bernoulli polynomials are defined, and its differential equation and recursion formulas are studied among other properties. It is well-known that some of the known properties of the Bernoulli numbers can be obtained from fundamental relationships between complete and elementary symmetric functions.…”