Related papers: Evaluations and relations for finite trigonometric…
We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were systematically first studied by Berndt…
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta…
A simple integration by parts and telescopic cancellation leads to a rigorous derivation of the first 2 terms for the error in Ramanujan's asymptotic series for the nth partial sum of the harmonic series. Then Kummer's transformation gives…
Riemann sums, a classical method for approximating the definite integral of a function, have been extensively studied in the past. However, their monotonic properties, while still of great importance, particularly in approximation theory…
We prove three theorems on finite real multiple zeta values: the symmetric formula, the sum formula and the height-one duality theorem. These are analogues of their counterparts on finite multiple zeta values.
We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (\pi n \tau)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $\tau$ a real quadratic…
By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined. Several explicit summation formulae are established.
Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial…
This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between…
By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman's double $t$-values and Kaneko-Tsumura's double $T$-values, and…
I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard…
We provide numerical procedures for possibly best evaluating the sum of positive series. Our procedures are based on the application of a generalized version of Kummer's test.
In this paper, we obtain a weighted trigonometric summation formula which is an extension of the trigonometric summation formula by Grigor'yan, Lin and Yau \cite{GLY}.
Many $\mathbb{Q}$-linear relations exist between multiple zeta values, the most interesting of which are various weighted sum formulas. In this paper, we generalized these to Euler sums and some other variants of multiple zeta values by…
We describe an effective method for calculating certain infinite sums, generalizations of the classical Bernoulli polynomials. As shown by Edward Witten in his papers on two-dimensional gauge theories, the correlation functions of…
We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/\pi^2.
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…
We obtain new trigonometric identities, which are some product-to-sum type formulas for the higher derivative of the cotangent and cosecant functions. Further, from specializations of our formulas, we derive not only various known…
We introduce and study a ``level two'' analogue of finite multiple zeta values. We give conjectural bases of the space of finite Euler sums as well as that of usual finite multiple zeta values in terms of these newly defined elements. A…