395 research outputs found
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 satisfying the condition . Substituting
into the resulting recurrence equations produces the famous
recursions for rational approximations to , due to
Ap\'ery, as well as the known recursion for rational approximations to
. 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
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
Multiple -Zeta Values
We introduce a -analog of the multiple harmonic series commonly referred
to as multiple zeta values. The multiple -zeta values satisfy a -stuffle
multiplication rule analogous to the stuffle multiplication rule arising from
the series representation of ordinary multiple zeta values. Additionally,
multiple -zeta values can be viewed as special values of the multiple
-polylogarithm, which admits a multiple Jackson -integral representation
whose limiting case is the Drinfel'd simplex integral for the ordinary multiple
polylogarithm when . The multiple Jackson -integral representation for
multiple -zeta values leads to a second multiplication rule satisfied by
them, referred to as a -shuffle. Despite this, it appears that many
numerical relations satisfied by ordinary multiple zeta values have no
interesting -extension. For example, a suitable -analog of Broadhurst's
formula for , if one exists, is likely to be rather
complicated. Nevertheless, we show that a number of infinite classes of
relations, including Hoffman's partition identities, Ohno's cyclic sum
identities, Granville's sum formula, Euler's convolution formula, Ohno's
generalized duality relation, and the derivation relations of Ihara and Kaneko
extend to multiple -zeta values.Comment: 35 page
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
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