Related papers: A New Generating Function for Calculating the Igus…

The representation zeta function of a profinite group $G$ encodes the distribution of continuous irreducible complex representations of $G$ as a function of the dimension. Its abscissa of convergence $\alpha(G)$ describes the polynomial…

In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork's unit root zeta function attached to an ordinary family of algebraic varieties defined over a…

The aim of this paper is to construct generating functions for new families of special polynomials including the Appel polynomials, the Hermite-Kamp\`e de F\`eriet polynomials, the Milne-Thomson type polynomials, parametric kinds of Apostol…

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of…

An alternative generating function is proposed to enumerate row-convex polyominoes without internal holes on a discrete grid. The approach is based on integer partitions of the total area, where each partition corresponds to a sequence of…

Let $f(t)=\sum_{n=0}^{+\infty}\frac{C_{f,n}}{n!}t^n$ be an analytic function at $0$, and let $C_{f, n}(x)=\sum_{k=0}^{n}\binom{n}{k}C_{f,k} x^{n-k}$ be the sequence of Appell polynomials, referred to as $\textit{C-polynomials associated to…

We are studing Galois actions on fundamental groups. Using towers of coverings we construct measures on the products of finite copies of Z_p. Using these measures we can calculate coefficients of Galois representations. In the simplest case…

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an…

The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study…

We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

We give an anecdotal discussion of the problem of searching for polynomials with all roots on the unit circle, whose coefficients are rational numbers subject to certain congruence conditions. We illustrate with an example from a…

For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalization of a conjecture of Igusa - Denef - Loeser predicts that every pole of its topological zeta function is a root of its Bernstein-Sato polynomial. However, typically…

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of $\z$-functions associated to conic manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic extensions…

Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let…

We prove a closed formula for the generating function $\mathsf Z_d(t)$ of the motives $[\mathrm{Hilb}^d(\mathbb A^n)_0] \in K_0(\mathrm{Var}_{\mathbb C})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an…

Let $p$ be a prime, and let $\Gamma=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic…

This paper completes the construction of $p$-adic $L$-functions for unitary groups. More precisely, in 2006, the last three named authors proposed an approach to constructing such $p$-adic $L$-functions (Part I). Building on more recent…

We study functionals of the form \[\zeta_{t}=\int_0^{t}...\int_0^{t} | X_1(s_1)+...+ X_p(s_p)|^{-\sigma}ds_1... ds_p\] where $X_1(t),..., X_p(t)$ are i.i.d. $d$-dimensional symmetric stable processes of index $0<\bb\le 2$. We obtain results…