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http://dx.doi.org/10.4134/JKMS.j180227

LEHMER'S GENERALIZED EULER NUMBERS IN HYPERGEOMETRIC FUNCTIONS  

Barman, Rupam (Department of Mathematics Indian Institute of Technology Guwahati)
Komatsu, Takao (School of Mathematics and Statistics Wuhan University)
Publication Information
Journal of the Korean Mathematical Society / v.56, no.2, 2019 , pp. 485-505 More about this Journal
Abstract
In 1935, D. H. Lehmer introduced and investigated generalized Euler numbers $W_n$, defined by $${\frac{3}{e^t+e^{wt}e^{w^2t}}}={\sum\limits_{n=0}^{\infty}}W_n{\frac{t^n}{n!}}$$, 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;
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1 F. Brioschi, Sulle funzioni Bernoulliane ed Euleriane, Annali de Mat., i. (1858), 260- 263; Opere Mat., i. pp. 343-347.   DOI
2 P. J. Cameron, Some sequences of integers, Discrete Math. 75 (1989), no. 1-3, 89-102.   DOI
3 J. W. L. Glaisher, Expressions for Laplace's coefficients, Bernoullian and Eulerian numbers etc. as determinants, Messenger (2) 6 (1875), 49-63.
4 A. Hassen and H. D. Nguyen, Hypergeometric Bernoulli polynomials and Appell sequences, Int. J. Number Theory 4 (2008), no. 5, 767-774.   DOI
5 A. Hassen and H. D. Nguyen, Hypergeometric zeta functions, Int. J. Number Theory 6 (2010), no. 1, 99-126.   DOI
6 F. T. Howard, A sequence of numbers related to the exponential function, Duke Math. J. 34 (1967), 599-615.   DOI
7 F. T. Howard, Some sequences of rational numbers related to the exponential function, Duke Math. J. 34 (1967), 701-716.   DOI
8 S. Hu and M.-S. Kim, On hypergeometric Bernoulli numbers and polynomials, Acta Math. Hungar. 154 (2018), no. 1, 134-146.   DOI
9 K. Kamano, Sums of products of hypergeometric Bernoulli numbers, J. Number Theory 130 (2010), no. 10, 2259-2271.   DOI
10 T. Komatsu, Incomplete poly-Cauchy numbers, Monatsh. Math. 180 (2016), no. 2, 271- 288.   DOI
11 T. Komatsu, Complementary Euler numbers, Period. Math. Hungar. 75 (2017), no. 2, 302-314.   DOI
12 T. Komatsu, Incomplete multi-poly-Bernoulli numbers and multiple zeta values, Bull. Malays. Math. Sci. Soc. 41 (2018), 2029-2040.   DOI
13 T. Komatsu, K. Liptai, and I. Mezo, Incomplete Cauchy numbers, Acta Math. Hungar. 149 (2016), no. 2, 306-323.   DOI
14 T. Komatsu and Y. Ohno, Lehmer's generalized Euler numbers, preprint.
15 T. Komatsu and J. L. Ramirez, Some determinants involving incomplete Fubini numbers, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 26 (2018), no. 3, 143-170.
16 T. Komatsu and H. Zhu, Hypergeometric Euler numbers, preprint. arXiv:1612.06210.
17 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Ann. of Math. (2) 36 (1935), no. 3, 637-649.   DOI
18 T. Muir, The Theory of Determinants in the Historical Order of Development, Four volumes, Dover Publications, New York, 1960.
19 N. J. A. Sloane, The on-line encyclopedia of integer sequences, available at oeis.org.
20 N. Trudi, Intorno ad alcune formole di sviluppo, Rendic. dell' Accad. Napoli (1862), 135-143.