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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.
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참고문헌 (20)
참고문헌 신청
F. Brioschi / 1858 / Sulle funzioni Bernoulliane ed Euleriane / Annali de Mat :260~263
P. J. Cameron / 1989 / Some sequences of integers / Discrete Math 75(1-3):89~102
J. W. L. Glaisher / 1875 / Expressions for Laplace's coe cients, Bernoullian and Eulerian numbers etc. as determinants / Messenger 6:49~63
A. Hassen / 2008 / Hypergeometric Bernoulli polynomials and Appell se-quences / Int. J. Number Theory 4(5):767~774
A. Hassen / 2010 / Hypergeometric zeta functions / Int. J. Number Theory 6(1):99~126
P. J. Cameron / 1989 / Some sequences of integers / Discrete Math 75(1-3):89~102
J. W. L. Glaisher / 1875 / Expressions for Laplace's coe cients, Bernoullian and Eulerian numbers etc. as determinants / Messenger 6:49~63
A. Hassen / 2008 / Hypergeometric Bernoulli polynomials and Appell se-quences / Int. J. Number Theory 4(5):767~774
A. Hassen / 2010 / Hypergeometric zeta functions / Int. J. Number Theory 6(1):99~126
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