This article is cited in 24 papers

Cyclic graphs and Apéry's theorem

V. N. Sorokin

M. V. Lomonosov Moscow State University

Abstract: This is a survey of results about the behaviour of Hermite–Padé approximants for graphs of Markov functions, and a survey of interpolation problems leading to Apéry's result about the irrationality of the value $\zeta(3)$ of the Riemann zeta function. The first example is given of a cyclic graph for which the Hermite–Padé problem leads to Apéry's theorem. Explicit formulae for solutions are obtained, namely, Rodrigues' formulae and integral representations. The asymptotic behaviour of the approximants is studied, and recurrence formulae are found.

UDC: 517.53

MSC: Primary 11M06, 11J72, 41A21, 05C90; Secondary 11J82, 14G10

Received: 15.03.2001

DOI: 10.4213/rm512

 English version: Russian Mathematical Surveys, 2002, 57:3, 535–571

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