Related papers: Simultaneous generation for zeta values by the Mar…
Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove…
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).…
Through a systematic approach on generating Wilf-Zeilberger-pairs, we prove some hypergeometric identities conjectures due to Z.W. Sun, J. Guillera and Y. Zhao etc., including two Ramanujan-$1/\pi^4$, one $1/\pi^3$ formulas as well as a…
We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental…
Plouffe conjectured rapidly converging series formulas for $\pi^{2n+1}$ and $\zeta (2n+1)$ for small values of $n$. We find the general pattern for all nonnegative integer values of $n$ and offer a proof.
We state and prove product formulae for several generating functions for sequences $(a_n)_{n\ge0}$ that are defined by the property that $Pa_n+b^2$ is a square, where $P$ and $b$ are given integers. In particular, we prove corresponding…
Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.
New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a normalized…
In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated…
For a prime $p$ and nonnegative integers $j$ and $n$ let $\vartheta_p(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are exactly divisible by $p^j$. Moreover, for a finite sequence $w=(w_{r-1}\cdots w_0)\neq…
I consider the Lerch-Hurwitz or periodic zeta function as covariance function of a periodic continuous-time stationary stochastic process. The function can be parametrized with a continuous index $\nu$ which regulates the continuity and…
We study the number of $k \times r$ plane partitions, weighted on the sum of the first row. Using Erhart reciprocity, we prove an identity for the generating function. For the special case $k=1$ this result follows from the classical theory…
We evaluate in closed form three special classes of alternating zeta series with one and two additional parameters. Two classes are expressed as linear combinations of polylogarithms while for the third class we prove an expression…
We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results…
A conjectural formula for the $k$-point generating function of Gromov--Witten invariants of the Riemann sphere for all genera and all degrees was proposed in \cite{DY2}. In this paper, we give a proof of this formula together with an…
We present explicit formulas for Hecke eigenforms as linear combinations of q-analogues of modified double zeta values. As an application, we obtain period polynomial relations and sum formulas for these modified double zeta values. These…
We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the Grothendieck ring of finite G-sets tensored by the field of rational numbers. Main two ingredients of the…
We study compositions of a positive integer $n$ in which the occurrence of even parts larger than a fixed threshold $k$ is controlled. More precisely, for each composition $m=(m_1,\dots,m_r)$ we consider the number of even parts strictly…
We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series…
We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product which is regarded as the Euler…