59,118 research outputs found
On simultaneous diophantine approximations to and
We present a hypergeometric construction of rational approximations to
and which allows one to demonstrate simultaneously the
irrationality of each of the zeta values, as well as to estimate from below
certain linear forms in 1, and with rational
coefficients. A new notion of (simultaneous) diophantine exponent is introduced
to formalise the arithmetic structure of these specific linear forms. Finally,
the properties of this newer concept are studied and linked to the classical
irrationality exponent and its generalisations given recently by S. Fischler.Comment: 23 pages; v2: new subsection 4.5 adde
Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]
Roger Apery's seminal method for proving irrationality is "turned on its
head" and taught to computers, enabling a one second redux of the original
proof of zeta(3), and many new irrationality proofs of many new constants,
alas, none of them is both famous and not-yet-proved-irrational.Comment: 16 pages. Exclusively published in the Personal Journal of Shalosh B.
Ekhad and Doron Zeilberger, May 2014, and this arxiv.org. Accompanied my
Maple package NesApery, available from
http://www.math.rutgers.edu/~zeilberg/tokhniot/NesAper
Continued fractions of certain Mahler functions
We investigate the continued fraction expansion of the infinite products
where polynomials satisfy
and . We construct relations between partial quotients of
which can be used to get recurrent formulae for them. We provide that
formulae for the cases and . As an application, we prove that for
where is an arbitrary rational number except 0 and 1, and for
any integer with such that the irrationality exponent
of equals two. In the case we provide a partial analogue of the
last result with several collections of polynomials giving the
irrationality exponent of strictly bigger than two.Comment: 25 page
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