Related papers: Sun's Series via Cyclotomic Multiple Zeta Values
We present a new "integral=series" type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear…
Using some transformation formulas of the generalized hypergeometric series $\,_3F_2$, we give another proof of D. Zagier's evaluation formula of the multiple zeta values $\zeta(2,...,2,3,2,...,2)$.
Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the…
This is an expansion of a talk for mathematics and physics students of the Manchester Grammar and Manchester High Schools. It deals with numbers such as the Riemann zeta value zeta(3)=sum_{n>0}1/n^3. Zeta values appear in the description of…
In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a…
Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings $(\{2\}^a,c,\{2\}^b)$ that appeared to be useful for proving new congruences for MHS as well as new…
In [Ok] Okounkov studies a specific $q$-analogue of multiple zeta values and makes some conjectures on their algebraic structure. In this note we compare Okounkovs $q$-analogues to the generating function for multiple divisor sums defined…
Inspired by the theory of Hodge correlators due to Goncharov and by the plectic principle of Nekov\'a\v{r} and Scholl, we construct higher plectic Green functions and give a higher order generalization of Hecke's formula for abelian…
We prove that any Mordell-Tornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any Mordell-Tornheim sum…
The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]^N…
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in…
In this note we introduce multi-interpolated multiple zeta values. We provide a basic decomposition of these objects involving ordered partitions. We also obtain identities for special instances of multi-interpolated multiple zeta values…
The cyclic sum formulas for multiple zeta and zeta-star values were respectively proved by Hoffman and Ohno, and Ohno and Wakabayashi. Kawasaki and Oyama obtained an analogous formulas for finite multiple zeta and zeta-star values. In this…
We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels…
In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$ \sum_{k=(p+1)/2}^{p-1}\frac{\binom{2k}k^2}{k16^k}\equiv-\frac{21}2H_{p-1}\pmod{p^4}, $$ where $H_n$ denotes the $n$-th harmonic…
In this article, we express solutions of the Gauss hypergeometric equation as a series of the multiple polylogarithms by using iterated integral. This representation is the most simple case of a semisimple representation of solutions of the…
For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…
Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0 \right\}}_r},1} \right)$. In this paper we show that the sums of multiple…
Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincar\'e series (zeta function).…
Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including…