175 research outputs found
An Ap\'ery-like difference equation for Catalan's constant
Applying Zeilberger's algorithm of creative telescoping to a family of
certain very-well-poised hypergeometric series involving linear forms in
Catalan's constant with rational coefficients, we obtain a second-order
difference equation for these forms and their coefficients. As a consequence we
obtain a new way of fast calculation of Catalan's constant as well as a new
continued-fraction expansion for it. Similar arguments can be put forward to
indicate a second-order difference equation and a new continued fraction for
, and we announce corresponding results at the end of this
paper.Comment: 10 pages; updating references (28 October 2002
One of the Odd Zeta Values from to Is Irrational. By Elementary Means
Available proofs of result of the type 'at least one of the odd zeta values
is irrational' make use of the saddle-point
method or of linear independence criteria, or both. These two remarkable
techniques are however counted as highly non-elementary, therefore leaving the
partial irrationality result inaccessible to general mathematics audience in
all its glory. Here we modify the original construction of linear forms in odd
zeta values to produce, for the first time, an elementary proof of such a
result - a proof whose technical ingredients are limited to the prime number
theorem and Stirling's approximation formula for the factorial
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds
We examine instances of modularity of (rigid) Calabi-Yau manifolds whose
periods are expressed in terms of hypergeometric functions. The -th
coefficients of the corresponding modular form can be often read off, at
least conjecturally, from the truncated partial sums of the underlying
hypergeometric series modulo a power of and from Weil's general bounds
, where is the weight of the form. Furthermore, the
critical -values of the modular form are predicted to be -proportional to the values of a related basis of solutions to the
hypergeometric differential equation
Irrationality of values of zeta-function
We present several results on the number of irrational and linear independent
values among , where is an odd
integer and is an integer. The main tool in our proofs is a certain
generalization of Rivoal's construction (math.NT/0008051, math.NT/0104221).Comment: 8+8 pages (English+Russian); to appear in the Proceedings of the
Conference of Young Scientists (Moscow University, April 9-14, 2001
- …