On Infinitely Many Rational Approximants to ζ(3)

Jorge Arvesú,Anier Soria-Lorente

doi:10.3390/math7121176

Jorge Arvesú, Anier Soria-Lorente

Open Access

https://doi.org/10.3390/math7121176

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Abstract

Highlights

A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems
In [7] the proof is addressed from the perspective of Hermite-Padé approximation problem involving multiple orthogonal polynomials [8,9,10], but that proof was not connected with Equation (1)
Mathematics 2019, 7, 1176 linear difference equation depend on other parameters as well as n

Summary

Introduction

A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ (3). A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. The holonomic character of Equation (36) leads to infinitely many sequences of rational approximants to ζ (3). This is because the holonomic sequences that are produced depend on up to three integer parameters (almost freely selected). The use of this technique is combinatorially more complicated when dealing with holonomic sequences that depend on several parameters

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Conclusion