Lehmer's generalized Euler numbers in hypergeometric functions
J. Korean Math. Soc. 2019 Vol. 56, No. 2, 485-505
Published online 2019 Mar 01
Rupam Barman, Takao Komatsu
Indian Institute of Technology Guwahati; Wuhan University
Abstract : In 1935, D. H. Lehmer introduced and investigated generalized Euler numbers $W_n$, defined by $$ \frac{3}{e^{t}+e^{\omega t}+e^{\omega^2 t}}=\sum_{n=0}^\infty W_n\frac{t^n}{n!}\,, $$ where $\omega$ is a complex root of $x^2+x+1=0$. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli and Euler numbers. These concepts can be generalized to the hypergeometric Bernoulli and Euler numbers by several authors, including Ohno and the second author. In this paper, we study more general numbers in terms of determinants, which involve Bernoulli, Euler and Lehmer's generalized Euler numbers. The motivations and backgrounds of the definition are in an operator related to Graph theory. We also give several expressions and identities by Trudi's and inversion formulae.
Keywords : Euler numbers, generalized Euler numbers, determinants, recurrence relations, hypergeometric functions
MSC numbers : Primary 11B68, 11B37, 11C20, 15A15, 33C20
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