Extending Gaussian hypergeometric series to the pp-adic setting

We define a function which extends Gaussian hypergeometric series to the $p$-adic setting. This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes. We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series. These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas.Comment: Int. J. Number Theory, accepted for publicatio

Well-poised generation of Ap\'ery-like recursions

The idea to use classical hypergeometric series and, in particular, well-poised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several novelties in number theory and neighbouring areas of mathematics. Here we present a systematic approach to derive second-order polynomial recursions for approximations to some values of the Lerch zeta function, depending on the fixed (but not necessarily real) parameter $\alpha$ satisfying the condition $\Re(\alpha)<1$. Substituting $\alpha=0$ into the resulting recurrence equations produces the famous recursions for rational approximations to $\zeta(2)$, $\zeta(3)$ due to Ap\'ery, as well as the known recursion for rational approximations to $\zeta(4)$. Multiple integral representations for solutions of the constructed recurrences are also given.Comment: 8 pages; to appear in the Proceedings of the 7th OPSFA (Copenhagen, 18--22 August 2003

Approximations to -, di- and tri- logarithms

We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apery-like (polynomial) recursions.Comment: 11 pages, minor correction

Generalization of the Lee-O'Sullivan List Decoding for One-Point AG Codes

We generalize the list decoding algorithm for Hermitian codes proposed by Lee and O'Sullivan based on Gr\"obner bases to general one-point AG codes, under an assumption weaker than one used by Beelen and Brander. Our generalization enables us to apply the fast algorithm to compute a Gr\"obner basis of a module proposed by Lee and O'Sullivan, which was not possible in another generalization by Lax.Comment: article.cls, 14 pages, no figure. The order of authors was changed. To appear in Journal of Symbolic Computation. This is an extended journal paper version of our earlier conference paper arXiv:1201.624

An Euler-type formula for β(2n)\beta(2n) and closed-form expressions for a class of zeta series

In a recent work, Dancs and He found an Euler-type formula for $\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in comparison to $\zeta(2n)$, which is a rational multiple of $\pi^{2n}$. For the Dirichlet beta function, the things are `inverse': $\beta(2n+1)$ is a rational multiple of $\pi^{2n+1}$ and no closed-form expression is known for $\beta(2n)$. Here in this work, I modify the Dancs-He approach in order to derive an Euler-type formula for $\,\beta{(2n)}$, including $\,\beta{(2)} = G$, the Catalan's constant. I also convert the resulting series into zeta series, which yields new exact closed-form expressions for a class of zeta series involving $\,\beta{(2n)}$ and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured.Comment: 11 pages, no figures. A few small corrections. ACCEPTED for publication in: Integral Transf. Special Functions (09/11/2011

Subresultants in multiple roots: an extremal case

We provide explicit formulae for the coefficients of the order-d polynomial subresultant of (x-\alpha)^m and (x-\beta)^n with respect to the set of Bernstein polynomials \{(x-\alpha)^j(x-\beta)^{d-j}, \, 0\le j\le d\}. They are given by hypergeometric expressions arising from determinants of binomial Hankel matrices.Comment: 18 pages, uses elsart. Revised version accepted for publication at Linear Algebra and its Application

Bounding the number of points on a curve using a generalization of Weierstrass semigroups

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup [P. Beelen, N. Tuta\c{s}: A generalization of the Weierstrass semigroup, J. Pure Appl. Algebra, 207(2), 2006] for an $n$-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in [O. Geil, R. Matsumoto: Bounding the number of $\mathbb{F}_q$-rational places in algebraic function fields using Weierstrass semigroups. Pure Appl. Algebra, 213(6), 2009]

Some remarks on the visible points of a lattice

We comment on the set of visible points of a lattice and its Fourier transform, thus continuing and generalizing previous work by Schroeder and Mosseri. A closed formula in terms of Dirichlet series is obtained for the Bragg part of the Fourier transform. We compare this calculation with the outcome of an optical Fourier transform of the visible points of the 2D square lattice.Comment: 9 pages, 3 eps-figures, 1 jpeg-figure; updated version; another article (by M. Baake, R. V. Moody and P. A. B. Pleasants) with the complete solution of the spectral problem will follow soon (see math.MG/9906132