Related papers: A curious identity for Bernoulli numbers
We find this identity, that looks like an exercise in Calculus 1, surprising, and beautiful. We hope that you would too.
The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhaber's well-known formula expressing the power sums as polynomials whose coefficients…
In this paper, we derive some new and interesting idebtities for Bernoulli, Euler and Hermite polynomials associated with Chebyshev polynomials.
The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernolli, degenerate…
Recently Zagier proved a remarkable q-series identity. We show that this identity can also be proved by modifying Franklin's classical proof of Euler's pentagonal number theorem.
We present a probabilistic proof of Euler's pentagonal number theorem based on a shuffling model.
The recent interest in $q$-Stirling numbers of the second kind in type B prompted us to give a type B analogue of a classical identity connecting the $q$-Stirling numbers of the second kind and Carlitz's major $q$-Eulerian numbers, which…
We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula.…
We continue our study on relationships between Bernoulli polynomials and balancing (Lucas-balancing) polynomials. From these polynomial relations, we deduce new combinatorial identities with Fibonacci (Lucas) and Bernoulli numbers.…
We prove formulas for the Bernoulli numbers by using the Newton-Girard identities to evaluate the Riemann zeta function at positive even integers. To do this, we define a sequence of positive integers, a sequence of polynomials, and a…
This note presents criteria in terms of Bernoulli numbers for a number to be simultaneously a Wilson prime and a Lerch prime.
Given two infinite sequences with known binomial transforms, we compute the binomial transform of the product sequence. Various identities are obtained and numerous examples are given involving sequences of special numbers: Harmonic…
In this note, we present several identities involving binomial coefficients and the two kind of Stirling numbers.
An identity for binomial symbols modulo an odd positive integer $n$ relating to the least prime factor of $n$ is proved. The identity is discussed within the context of Pell conics.
Binomial versions of the Andrews-Gordon-Bressoud identities are given.
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 this note, we provide a conceptual explanation of a well-known polynomial identity used in algebraic number theory.
We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers.
In this paper we present several natural $q$-analogues of the poly-Bernoulli numbers arising in combinatorial contexts. We also recall some relating analytical results and ask for combinatorial interpretations.
The classical sequence of Bernoulli numbers is known to the the sequence of moments of a family of orthogonal polynomials. Some similar statements are obtained for another sequence of rational numbers, which is similar in many ways to the…