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Legendre transforms and Apéry's sequences

Part of: Sequences and sets Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

Asmus L. Schmidt
Asmus L. Schmidt
Affiliation:
Matematisk Institut, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
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Abstract

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This article studies particular sequences satisfying polynomial recurrences, among those Apéry's sequence which is shown to be the Legendre transform of the sequence. This results in the construction of simultaneous approximations of π 2/8 and ζ(3).

MSC classification

Secondary: 11B37: Recurrences 11J13: Simultaneous homogeneous approximation, linear forms 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 3 , June 1995 , pp. 358 - 375
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Askey, R. and Wilson, J. A., ‘A recurrence relation generalizing those of Apéry’, J. Austral. Math. Soc. (Series A) 36 (1984), 267–278.CrossRefGoogle Scholar
[2]Cusick, T., ‘Recurrences for sums of powers of binomial coefficients’, J. Combin. Theory Ser. A 52 (1989), 77–83.CrossRefGoogle Scholar
[3]Franel, J., L'Intermédiaire des Mathématiciens vol. 1 (1894).Google Scholar
[4]Franel, J., L'Intermédiaire des Mathématiciens vol. 2 (1895).Google Scholar
[5]Perlstadt, M. A., ‘Some recurrences for sums of powers of binomial coefficients’, J. Number Theory 27 (1987), 304–309.CrossRefGoogle Scholar
[6]van der Poorten, A. J., ‘A proof that Euler missed…Apéry's proof of the irrationality of ζ(3)’, Math. Intelligencer 1 (1978/1979), 195–203.CrossRefGoogle Scholar
[7]Rademacher, H., Topics in analytic number theory (Springer, Berlin, 1973).CrossRefGoogle Scholar
[8]Riordan, J., Combinatorial identities (Wiley, New York, 1968).Google Scholar
[9]Schmidt, A. L., ‘Generalized Legendre polynomials’, J. Reine Angew. Math. 404 (1990), 192–202.Google Scholar
[10]Schmidt, A. L., ‘Generalized q-Legendre polynomials’, J. Comput. Appl. Math. 49 (1993), 243–249.CrossRefGoogle Scholar
[11]Slater, L. J., Generalized hypergeometric functions (Cambridge Univ. Press, London, 1966).Google Scholar
[12]Staver, T. B., ‘Om summasjon av potenser av binomialkoefficientene’, Norsk Matematisk Tidskrift 29 (1947), 97–103.Google Scholar
[13]Strehl, V., ‘Binomial identities — combinatorial and algorithmic aspects’, Discrete Math. 136 (1994), 309–346.CrossRefGoogle Scholar
[14]Szegö, G., Orthogonal polynomials (Amer. Math. Soc., Providence, 1975).Google Scholar