2-Adic Congruences of Nörlund Numbers and of Bernoulli Numbers of the Second Kind

Adelberg, Arnold

doi:10.1006/jnth.1998.2278

Cited by 9 publications

(20 citation statements)

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“…where c n = (−2) n b n − 1 for all n, as in [1]. We define rational integers c r,n by ∞ n=0 c r,n t n = 2 r+1 t (1 − 2t) 2 r + 1 (2.5) so that c n = ∞ r=1 c r,n as a convergent sum in Z 2 for all n 0.…”

Section: Proof Of 2-adic Formulamentioning

confidence: 99%

“…For n = w the numbers B (n) n are called Nörlund numbers [4], or Cauchy numbers of the second type [3,7], and may be determined by the generating function Adelberg also studied their 2-adic expansion in [1] and proved that (−2) n B (n) n /n! ≡ −1 (mod 2 [n/2]+1 Z 2 ), with this congruence being best possible if n ≡ 2 (mod 4) [1, Theorem 2 and Corollary 1 of §3]; he further conjectured "stable congruences" for them which were supported by numerical calculations.…”

Section: Introductionmentioning

confidence: 99%

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A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

Young

2008

Journal of Number Theory

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Section: Proof Of 2-adic Formulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

Young

2008

Journal of Number Theory

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“…. for Euler's constant, γ m for mth generalized Euler's constant (Stieltjes constant) accordingly to their definition (2), 5 ( k n ) denotes the binomial coefficient C n k , B n stands for the nth Bernoulli number, 6 H n and H (s) n denote the nth harmonic number and the nth generalized harmonic number of order s…”

Section: I2 Notations and Some Definitionsmentioning

confidence: 99%

“…For more information about these important coefficients, see [135, pp. 240-251], [136], [90, p. 132 [30,82,191,6,192,28], [124,Eq. (3)], [123,129], [9, pp.…”

Section: Iii1 Introductionmentioning

confidence: 99%

“…22 These numbers, called by some authors signless generalized Bernoulli numbers |B (n) n | and signless Nørlund numbers, are much less famous than Gregory's coefficients G n , but their study is also very interesting, see [135, pp. 150-151], [47, p. 12], [136], [13, vol [81,6,192,143,18]. 23 Strictly speaking, Binet found only first four coefficients of the corresponding series for the digamma function and incorrectly calculated the last coefficient (for K (5) [12] and Gerst [60].…”

Section: Iii1 Introductionmentioning

confidence: 99%

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Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

Blagouchine

2017

Journal of Number Theory

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In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γ m are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π −2 with rational coefficients and converges slightly better than Euler's series ∑ n −2 . The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.

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Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1

Blagouchine

2016

Journal of Mathematical Analysis and Applications

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In this paper, two new series for the logarithm of the Γ-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to π −1 . In particular, for any value of the form ln Γ( 1 2 n ± απ −1 ) and Ψ k ( 1 2 n ± απ −1 ), where Ψ k stands for the kth polygamma function, α is positive rational greater than 1 6 π, n is integer and k is non-negative integer, these series have rational terms only. In the specified zones of convergence, derived series converge uniformly at the same rate as ∑(n ln m n) −2 , where m = 1, 2, 3, . . . , depending on the order of the polygamma function. Explicit expansions into the series with rational coefficients are given for the most attracting values, such as ln. Besides, in this article, the reader will also find a number of other series involving Stirling numbers, Gregory's coefficients (logarithmic numbers, also known as Bernoulli numbers of the second kind), Cauchy numbers and generalized Bernoulli numbers. Finally, several estimations and full asymptotics for Gregory's coefficients, for Cauchy numbers, for certain generalized Bernoulli numbers and for certain sums with the Stirling numbers are obtained. In particular, these include sharp bounds for Gregory's coefficients and for the Cauchy numbers of the second kind.

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2-Adic Congruences of Nörlund Numbers and of Bernoulli Numbers of the Second Kind

Cited by 9 publications

References 5 publications

A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1

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