Hypergeometric Euler numbers

TL;DR

This paper introduces hypergeometric Euler numbers, explores their properties, expressions, and sums, and discusses their significance as natural extensions of classical Euler numbers with potential for further mathematical generalizations.

Contribution

It presents the definition, properties, and new generalizations of hypergeometric Euler numbers, expanding the understanding of Euler number analogues.

Findings

Defined hypergeometric Euler numbers and their complementary versions

Derived expressions and sums involving these numbers

Highlighted their importance as natural extensions of classical Euler numbers

Abstract

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors.