Related papers: Explicit formula for even-index Bernoulli numbers
In our previous publication we have shown a method for calculating series of even powers of $\pi$ based on the product representation of the $sinc$ function. We refer the readers to [1] for more details. In this work we apply the method to…
We define recursive harmonic numbers as a generalization of harmonic numbers. The table of recursive harmonic numbers, which is like Pascal's triangle, is constructed. A formula for recursive harmonic numbers containing binomial…
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the…
In 1938 E. T. Bell introduced "The Iterated Exponential Integers". He proved that these numbers may be expressed by polynomials with rational coefficients. However, Bell gave no formulas for any of the coefficients except the trivial one,…
We prove several Stern's type congruences for generalized bernoulli numbers.
In this paper we propose and solve a generalization of the Bernoulli Differential Equation, by means of a generalized fractional derivative. First we prove a generalization of Gronwall's inequality, which is useful for studying the…
Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann's zeta function. To do this, we explore properties of some…
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…
In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.
In this paper, we introduce poly-Bernoulli numbers with level $2$, related to the Stirling numbers of the second kind with level $2$, and study several properties of poly-Bernoulli numbers with level $2$ from their expressions, relations,…
Exact rational partitions are presented for Bernoulli and Euler numbers as novel sums involving Faulhaber and Sali\'e coefficients.
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the…
In this paper we give Kummer's original type congruence relation modulo a prime power for the universal Bernoulli numbers. Although the index of the power is half of original congruence, this estimate is best possible.
In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.
Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is…
We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
Let ``Faulhaber's formula'' refer to an expression for the sum of powers of integers written with terms in n(n+1)/2. Initially, the author used Faulhaber's formula to explain why odd Bernoulli numbers are equal to zero. Next, Cereceda gave…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{{\pi}^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $\zeta (n)$ is very near 1…