Related papers: On a generating function of Niebur-Poincar\'e seri…

Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic…

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and…

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by…

Let $\Gamma$ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$, and $\square$ the Laplacian which is a second-order hyperbolic differential operator. We…

It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed,…

Two methods of constructing 2D Toda $\tau$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group…

Let $\mathbb Z/d\mathbb Z \wr \mathfrak S_n$ denote the wreath product of the cyclic group $\mathbb Z/d\mathbb Z$ with the symmetric group $\mathfrak S_n$. We define generating functions for monomial (induced one-dimensional) characters of…

We prove a new generation result in $L^1$ for a large class of non-local operators with non-degenerate local terms. This class contains the operators appearing in Fokker-Planck or Kolmogorov forward equations associated with L\'evy driven…

The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our…

We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents…

We study the von Neumann algebra generated by q--deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q--deformed canonical commutation relations l_i l_j^*-q l_j^* l_i=delta_{ij} for -1<q<1. We show that if the number of…

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $\Gamma_0(2)$ of the modular group $SL_2(\mathbb{Z})$ is constructed. These nonlinear equations are analogues of the well known…

Fixing an arithmetic lattice $\Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ with $[\Gamma : \Gamma \cap \Delta] [\Delta: \Gamma \cap \Delta] =…

In this paper we analyze the quantum homological invariants (the Poincar\'e polynomials of the $\mathfrak{sl}_N$ link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the…

In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius-Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of…

A set of functions is defined which is indexed by a positive integer $n$ and partitions of integers. The case $n=1$ reproduces the standard Schur polynomials. These functions are seen to arise naturally as a determinant of an action on the…

Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key…

We investigate multi-graded Gorenstein semigroup algebras associated with an infinite family of reflexive lattice simplices. For each of these algebras, we prove that their multigraded Poincar\'e series is rational. Our method of proof is…

We decompose the generating function $\sum_{n=0}^\infty\binom{2n}nP_n(y)^2z^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of $\textit{second}$ order differential…

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential…