Related papers: On Some Summation Formulas
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
In this paper, we prove two results related to the solutions of norm form equations. Firstly, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations.…
We derive a formula connecting the orders of the automorphism groups of a finite group and of its covering groups.
In this paper, we introduce a new function computing the harmonic mean of element orders of a finite group. We present a series of properties for this function, and then we study groups for which the value of the function is an integer.
We provide a systematic formula, in terms of integer partitions, that generates perturbation theory explicitly at an arbitrary order. Our approach naturally includes an infinite number of perturbations and uses a single matrix equation that…
This paper provides bounds for the number of terms, denoted by $f$, of a harmonic sum with the condition that it starts from any arbitrary unit fraction $\frac{1}{m}$, $m > 1$, until another unit fraction $\frac{1}{m+f-1}$ such that the sum…
We present a partial proof of van Hoeij-Abramov conjecture about the algorithmic possibility of computation of finite sums of rational functions. The theoretical results proved in this paper provide an algorithm for computation of a large…
We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.
In this paper, we present the foundations of Summability Calculus, which places various established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special…
Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.
We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
The sum formula is one of the most well-known relations among multiple zeta values. This paper proves a conjecture of Kaneko predicting that an analogous formula holds for finite multiple zeta values.
In this short note we recall the definition of intrinsically harmonic forms, some known results and some open problems.
In this note some properties of the sum of element orders of a finite abelian group are studied.
Through a brute-force approach to calculating the higher derivatives of the falling factorial function, a number of interesting quantities were obtained and analyzed. In particular, it was found that a quantity that can be described as the…
We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.