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Multiple Legendre polynomials in diophantine approximation. (English) Zbl 1315.11064

The purpose of this paper is to extend the author’s results of [Acta Arith. 139, No. 2, 147–184 (2009; Zbl 1197.11083)] to obtain a lower bound of the irrationality measure of logarithms of rational numbers by algebraic numbers of bounded degree. The author introduces a class of multiple Legendre polynomials and functions as an application of the Rhin-Viola method, and obtains analytic and arithmetic properties of these polynomials and functions by means of a combinatorial identity involving hyperharmonic numbers, and proves that these polynomials and functions satisfy an Apéry-like recurrence. He gives new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree. He proves that the nonquadraticity exponent of \(\log 2\) is bounded from above by \(12.841618\dots\), improving a recent result of the author. His construction also yields some other known results in [C. Viola, Lond. Math. Soc. Lect. Note Ser. 247, 353–360 (1997; Zbl 0904.11020); with F. Amoroso, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No. 1, 225–249 (2001; Zbl 1008.11028); with the author, J. Aust. Math. Soc. 92, No. 2, 237–267 (2012; Zbl 1271.11076); A. Heimonen et al., Manuscr. Math. 81, No. 1–2, 183–202 (1993; Zbl 0801.11032)].

MSC:

11J82 Measures of irrationality and of transcendence
11J04 Homogeneous approximation to one number
11J17 Approximation by numbers from a fixed field
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[10] DOI: 10.4064/aa139-2-5 · Zbl 1197.11083
[11] DOI: 10.1017/S1446788712000274 · Zbl 1271.11076
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