## 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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### References:

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