Related papers: Non-recursive expressions for even-index Bernoulli…
Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives an explicit formula for Bernoulli numbers of even index. The formula contains a remarkable sequence of determinants.
We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…
We generalise the Bernoulli numbers to include the case where the index may be a continuous variable.
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the…
This paper shows that a finite discrete convolution involving Stirling numbers of both kinds and harmonic numbers can be expressed in terms of the Bernoulli numbers. As applications of this expression, the linear recurrence relation for the…
We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct Bernoulli denominators and some related problems.
In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare…
We derive an expression for the generalized Bernoulli numbers in terms of the Bernoulli numbers involving the (exponential) complete Bell polynomials.
In the paper, the authors establish an explicit formula for computing Bernoulli polynomials at non-negative integer points in terms of $r$-Stirling numbers of the second kind.
In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit…
In this paper, we give some recurrence formula and new and interesting identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus.
In the note, the author discovers an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.
We introduce generalized hypergeometric Bernoulli numbers for Dirichlet characters. We study their properties, including relations, expressions and determinants. At the end in Appendix we derive first few expressions of these numbers.
In the paper, the authors review some explicit formulas and establish a new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers of the second kind.
We prove a curious identity for the Bernoulli numbers.
We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain…
In the paper, the author finds an explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind.
In this paper, we investigate new class of sequences related to fully degenerate Bernoulli numbers and polynomials. From those sequences, we derive some formulae for the degenerate Bernoulli and Euler polynomials.
As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index have been established. For the multi-indexed…