Related papers: Summation formulas involving generalized harmonic …
We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic…
One-parameter generalizations of the logarithmic and exponential functions have been obtained as well as algebraic operators to retrieve extensivity. Analytical expressions for the successive applications of the sum or product operators on…
Two types of finite series of products of harmonic numbers involving nonnegative integer powers are evaluated, also yielding two other important harmonic number identities. The recursion formulas for these sums are derived, which are easily…
In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…
We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic…
We present here a large collection of harmonic and quadratic harmonic sums, that can be useful in applied questions, e.g., probabilistic ones. We find closed-form formulae, that we were not able to locate in the literature.
After reviewing some fundamental facts from the theory of theta hypergeometric series we derive, using indefinite summation, several summation, transformation, and expansion formulas for multibasic theta hypergeometric series. Some of the…
A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of…
In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.
In this study, depending on the upper and the lower indices of the hyperharmonic number $h_{n}^{(r)}$, nonlinear recurrence relations are obtained. It is shown that generalized harmonic number and hyperharmonic number can be obtained from…
We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…
We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct new summation formulas…
In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the…
A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and…
In terms of the telescoping method, a simple binomial sum is given. By applying the derivative operators to the equation just mentioned, we establish several general harmonic number identities including some known results.
In this paper we present some interesting results involving summation of series in particular trigonometric ones. We failed to locate these results in existing literature or in the web like MathWorld (http://mathworld.wolfram.com/) nor…
During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously…