Cyclic graphs and Apéry's theorem

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 ζ(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.