Hypergeometric 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.

Two types of hypergeometric degenerate Cauchy numbers

“…is the falling factorial with (x) 0 = 1. Similar hypergeometric numbers are hypergeometric Bernoulli numbers [9][10][11] and hypergeometric Euler numbers [12,13].…”

Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers

“…Komatsu and Zhu [17] introduced complementary Euler numbers, E n , as a special case of hypergeometric Euler numbers (see also [14]) by the generating function…”

Cameron's operator in terms of determinants, and hypergeometric numbers

“…When N = 0, E * n,≤m = E * 0,n,≤m are modified restricted Euler numbers of the second kind. When m → ∞, E N,n = E * N,n,≤∞ are the original hypergeometric Euler numbers of the second kind ( [19,27]) defined by the generating function…”

“…When n ≤ m − 1 for A * n,≤m = E * N,n,≤m in Proposition 2 or when m = 1 for A * n,≥m = E * N,n,≥m in Proposition 3, the result is reduced to a determinant expression of hypergeometric Euler numbers of the second kind ( [27]). In addition, when N = 1, we have a determinant expression of Euler numbers of the second kind ( [19,27]).…”