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.
Highlights
Euler numbers En are defined by the generating function 1 cosh t = ∞ n=0 tn En n! (1)One of the different definitions is et 2 +(see e.g. [3])
2 Determinant expressions of hypergeometric numbers. These 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
By using Proposition 1 or the relation (4), we have a determinant expression of hypergeometric Euler numbers ([13])
Summary
One kind of poly-Euler numbers is a typical generalization, in the aspect of L-functions ([21, 22, 23]). A different type of generalization is based upon hypergeometric functions. When N = 1, Bn = B1,n are classical Bernoulli numbers defined by et t −. When N = 1, cn = c1,n are classical Cauchy numbers defined by t log(1 +. When N = 0, En = E0,n are classical Euler numbers defined in (1). To poly-Euler numbers ([21, 22, 23]), hypergeometric Euler numbers are rational numbers, though the classical Euler numbers are integers. ). We record the first few values of EN,n: EN,.
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