Related papers: Finite sums associated with some polynomial identi…
We consider the reflection identities for harmonic sums at weight four. We decompose a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or positive values of the…
In this note we present a method for obtaining a wide class of combinatorial identities. We give several examples, in particular, based on the Gamma and Beta functions. Some of them have already been considered by previously, and other are…
An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits…
In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series…
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study…
A recent paper of A. Sofo proves some results about sums of products of quadratic alternating harmonic numbers and reciprocal binomial coefficients. In this paper, we extend his result to cubic alternating harmonic number sums and develop…
Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. For p; q to coprime positive integers and r we consider the finite trigonometric sums involving the product of three trigonometric…
We discuss various aspects of representation of a polynomial as a sum of monomials (for example, uniqueness of such representation and related estimations).
A new sums-of-tails identity involving two parameters $b$ and $d$ is obtained and is used to derive more results of similar type. One of Ramanujan's sums-of-tails identities from the Lost Notebook is shown to be a special case of our…
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
In this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials.
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example,…
Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of $2\times 2$ minors of certain recursive matrices, the alternating sums of their $2\times 2$ minors, and the sums…
Recent advances in our understanding of higher derived limits carry multiple implications in the fields of condensed and pyknotic mathematics, as well as for the study of strong homology. These implications are thematically diverse,…
A generalization of the Chu-Vandermonde convolution is presented and proved with the integral representation method. This identity can be transformed into another identity, which has as special cases two known identities. Another identity…
In this paper we study the possibility to define irreducible representations of the symmetric groups with the help of finitely many relations. The existence of finite bases is established for the classes of representations corresponding to…
We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…
This paper contains a number of series whose coefficients are products of central binomial coefficients & harmonic numbers. An elegant sum involving $\zeta(2)$ and two other nice sums appear in the last section.