Related papers: On an exponential power sum
We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…
This paper introduces a symbolic calculus-based approach for deriving closed-form expressions for the sums of arithmetic sequences. The method extends beyond constant-difference sequences to those with polynomially increasing steps,…
The n-th Delannoy number and the n-th Schr\"oder number given by $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ and $S_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an…
In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…
In this paper we continue to investigate the properties of those sequences $\{a_n\}$ satisfying the condition $\sum_{k=0}^n\binom nk(-1)^ka_k=\pm a_n$ $(n\ge 0)$. As applications we deduce new recurrence relations and congruences for…
Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite…
The study of degenerate versions of certain special polynomials and numbers, which was initiated by Carlitz's work on degenerate Euler and degenerate Bernoulli polynomials, has recently seen renewed interest among mathematicians. The aim of…
This translation has been published in Stephen Hawking (ed.), "God Created the Integers", published in 2007 by Running Press. There may have been some changes to the final published version and this copy. This is a translation from the…
In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as…
We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…
In this paper, we consider sums of values of degenerate falling factorials and give a probabilistic proof of a recurrence relation for them. This may be viewed as a degenerate version of the recent probabilistic proofs on sums of powers of…
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial…
The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on…
We realize that geometric polynomials and p-Bernoulli polynomials and numbers are closely related with an integral representation. Therefore, using geometric polynomials, we extend some properties of Bernoulli polynomials and numbers such…
In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.
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.
In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.
A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the…