Related papers: Certain character sums and hypergeometric series

In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric…

We study characters of states in $p$-adic vertex operator algebras. In particular, we show that the image of the character map for both the $p$-adic Heisenberg and $p$-adic lattice vertex operator algebras contain infinitely-many…

Explicit expressions for the hypergeometric series ${}_2F_1(-n, a; 2a\pm j;2)$ and ${}_2F_1(-n, a; -2n\pm j;2)$ for positive integer $n$ and arbitrary integer $j$ are obtained with the help of generalizations of Kummer's second and third…

With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by…

Let $p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as…

Fourier transformations of several functions of one and two variables are evaluated and then used to derive some integral and series identities. It is shown that certain double Mordell integrals can be reduced to a sum of products of…

In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet's result involving two $_8\psi_8$ series and two $_8\phi_7$ series.

The Clausen's Hypergeometric Function is given by $${}_3F_2(a,b,c;d,e;z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n(c)_n}{(d)_n(e)_n(1)_n}z^n\, ; \ a,b,c,d,e\in \mathbb{C}$$ provided $d,\, e\, \neq 0,-1,-2,\cdots$ in unit disc $\mathbb{D} =\{z\in…

In this article we study $p$-adic properties of sequences of integers (or $p$-adic integers) that satisfy a linear recurrence with constant coefficients. For such a sequence, we give an explicit approximate twisted interpolation to $\mathbb…

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument.…

We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit…

We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic…

Some aspects of analysis involving fields with absolute value functions are discussed, which includes the real or complex numbers with their standard absolute values, as well as ultrametric situations like the p-adic numbers.

Multidimensional matrix inversions provide a powerful tool for studying multiple hypergeometric series. In order to extend this technique to elliptic hypergeometric series, we present three new multidimensional matrix inversions. As…

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute…

We calculate some finite and infinite sums containing the digamma function in closed-form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative…

We present results for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-dimensional and two-dimensional series. Most of these series can be expressed in…

The weight two Eisenstein series may be considered as the first example of a Katz $p$-adic modular form. Classically, its values are defined for the primes of ordinary reduction. We offer a modified definition which applies uniformly to all…

We establish four supercongruences between truncated ${}_3F_2$ hypergeometric series involving $p$-adic Gamma functions, which extend some of the Rodriguez-Villegas supercongruences.

In an earlier preprint (math.AG/9810142) we gave an explicit description of the algebraic closure of the field of power series over a field of characteristic p, in terms of "generalized power series". In this paper, we give an analogous…