Results 1 to 10 of about 100,985 (159)
Bernoulli numbers and solitons [PDF]
Journal of Nonlinear Mathematical Physics, 2005 We present a new formula for the Bernoulli numbers as the following integral $$B_{2m} =\frac{(-1)^{m-1}}{2^{2m+1}} \int_{-\infty}^{+\infty} (\frac{d^{m-1}}{dx^{m-1}} {sech}^2 x)^2dx. $$ This formula is motivated by the results of Fairlie and Veselov, who
Grosset, M-P., Veselov, A. P.
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Explicit Formulas Involving -Euler Numbers and Polynomials [PDF]
Abstract and Applied Analysis, 2012 We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator.
Serkan Araci +2 more
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Probabilistic poly-Bernoulli numbers
Mathematical and Computer Modelling of Dynamical SystemsAssume that is Y a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to study probabilistic poly-Bernoulli numbers associated with Y, as probabilistic extensions of poly-Bernoulli numbers ...
Wencong Liu +3 more
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A note on polyexponential and unipoly Bernoulli polynomials of the second kind
Open Mathematics, 2021 In this paper, the authors study the poly-Bernoulli numbers of the second kind, which are defined by using polyexponential functions introduced by Kims. Also by using unipoly function, we study the unipoly Bernoulli numbers of the second kind, which are ...
Ma Minyoung, Lim Dongkyu
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Advances in Difference Equations, 2021 Recently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli
Waseem A. Khan +3 more
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Fully degenerate Bernoulli numbers and polynomials
Demonstratio Mathematica, 2022 The aim of this article is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Zp{
Kim Taekyun, Kim Dae San, Park Jin-Woo
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Several properties of hypergeometric Bernoulli numbers
Journal of Inequalities and Applications, 2019 In this paper, we give several characteristics of hypergeometric Bernoulli numbers, including several identities for hypergeometric Bernoulli numbers which the convergents of the continued fraction expansion of the generating function of the ...
Miho Aoki +2 more
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Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function
Advances in Difference Equations, 2021 A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function.
Taekyun Kim +4 more
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Congruences among generalized Bernoulli numbers [PDF]
Acta Arithmetica, 1995 Let \(\chi\) denote a primitive quadratic character mod \(M\) (or the trivial character) and let \(d\) be a fundamental discriminant (or 1). Denote by \(\chi'\) the character mod \(M |d |\) induced by \(\chi\). The authors consider the generalized Bernoulli numbers \(B_{m, \chi'}\) and the corresponding Bernoulli polynomials \(B_{m, \chi'} (X)\) at \(X
Szmidt, J. +2 more
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Extended Bernoulli and Stirling matrices and related combinatorial identities [PDF]
, 2013 In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers ...
Can, Mümün, Dağlı, M. Cihat
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