# A Third-Order Apéry-Like Recursion for ζ(5)

@article{Zudilin2002ATA,
title={A Third-Order Ap{\'e}ry-Like Recursion for $\zeta$(5)},
author={V. V. Zudilin},
journal={Mathematical Notes},
year={2002},
volume={72},
pages={733-737}
}
In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, a similar second-order difference equation for $\zeta(4)$ has been discovered. The note contains a possible generalization of the above results for the number $\zeta… Expand 22 Citations A single parameter Hermite–Padé series representation for Apéry’s constant • Mathematics • 2018 Inspired by the results of Rhin and Viola (2001), the purpose of this work is to elaborate on a series representation for$\zeta \left( 3\right)$which only depends on one single integer parameter.Expand Recurrence Relations and the Algebraic Irrationality of Ζ(3) We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, we show that there exists a four-term real linear recurrenceExpand On Zudilin-like rational approximations to ζ(5) In this paper we obtain two Zudilin-Like recurrence relations of third order for ζ(5), after applying Zeilberger’s algorithm of creative telescoping to some hypergeometric series. These recurrenceExpand Well-poised generation of Apéry-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 inExpand N T ] 2 8 N ov 2 00 3 WELL-POISED GENERATION OF APÉRY-LIKE RECURSIONS Abstract. The idea to use classical hypergeometric series and, in particular, wellpoised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several noveltiesExpand On Catalan’S constant A new efficient construction of Diophantine approximations to Catalan’s constant is presented that is based on the direct analysis of the representation of a hypergeometric function with speciallyExpand Rational approximations for values of derivatives of the Gamma function The arithmetic nature of Euler's constant γ is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence withExpand Mirror Maps and Zeta Values The aim of this work is an analytic investigation of differential equations producing mirror maps as well as giving new examples of mirror maps; one of these examples is related to (rationalExpand Hypergeometric approximations to polylogarithms We present three applications of the hypergeometric method to the arithmetic study of polylogarithm values and, in particular, of zeta values. Part 1 is joint work with Khodabakhsh and TatianaExpand Principal solutions of recurrence relations and irrationality questions in number theory We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of three-termExpand #### References SHOWING 1-10 OF 22 REFERENCES A Note on the Irrationality of ζ(2) and ζ(3) At the “Journees Arithmetiques” held at Marseille-Luminy in June 1978, R. Apery confronted his audience with a miraculous proof for the irrationality of ζ(3) = l-3+ 2-3+ 3-3 + .... The proof wasExpand An elementary proof of Apery's theorem We present a new elementary' proof of the irrationality of$\zeta(3)\$ based on some recent hypergeometric' ideas of Yu.Nesterenko, T.Rivoal, and K.Ball, and on Zeilberger's algorithm of creativeExpand
Well-poised hypergeometric service for diophantine problems of zeta values | NOVA. The University of Newcastle's Digital Repository
On montre comment les concepts classiques de series et integrales hypergeometriques bien equilibrees devient crucial dans l'etude des proprietes arithmetiques des valeurs de la fonction zeta deExpand
Apéry-like difference equation for Catalan's constant, E-print math
• Apéry-like difference equation for Catalan's constant, E-print math
• 2002
Nauk [Russian Math. Surveys]
• Nauk [Russian Math. Surveys]
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One algorithm for fast calculation of ζ(4) [in Russian
• Preprint, M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
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One algorithm for fast calculation of ζ(4) [in Russian], Preprint, M
• V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow,
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One algorithm for fast calculation of ζ(4), Preprint
• Russian Academy of Sciences, M. V. Keldysh Institute for Applied Mathematics
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Zudilin), in: E-print math
• Zudilin), in: E-print math
• 2002