Several properties of hypergeometric Bernoulli numbers

TL;DR

This paper explores properties of hypergeometric Bernoulli numbers, providing determinant formulas, relations to classical Bernoulli numbers, and explicit forms of their generating function convergents.

Contribution

It introduces new determinant expressions, inversion relations, and explicit generating function forms for hypergeometric Bernoulli numbers, expanding understanding of their structure.

Findings

Determinant expressions for hypergeometric Bernoulli numbers

Relations including Kummer's congruences between hypergeometric and classical Bernoulli numbers

Explicit forms of convergents of the generating function

Abstract

In this paper, we give the determinant expressions of the hypergeometric Bernoulli numbers, and some relations between the hypergeometric and the classical Bernoulli numbers which include Kummer's congruences. By applying Trudi's formula, we have some different expressions and inversion relations. We also determine explicit forms of convergents of the generating function of the hypergeometric Bernoulli numbers, from which several identities for hypergeometric Bernoulli numbers are given.