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Diagonal convergence of the remainder Padé approximants for the Hurwitz zeta function. (English) Zbl 07318741
Summary: The Hurwitz zeta function \(\zeta(s,a)\) admits a well-known (divergent) asymptotic expansion in powers of \(1/a\) involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Padé approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers \(\geq 2\). The latter can be viewed, in a certain sense, as analogues of Apéry’s celebrated sequences of rational approximations to \(\zeta(2)\) and \(\zeta(3)\).
11Y60 Evaluation of number-theoretic constants
41A21 Padé approximation
11J72 Irrationality; linear independence over a field
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI
[1] Apéry, R., Irrationality of \(\zeta(2)\) and \(\zeta(3)\), Astérisque, 61, 11-13 (1979) · Zbl 0401.10049
[2] Askey, R.; Wilson, J., A set of hypergeometric orthogonal polynomials, SIAM J. Math. Anal., 13, 4, 651-655 (1982) · Zbl 0496.33007
[3] Brezinski, C., Padé-Type Approximation and General Orthogonal Polynomials, International Series of Numerical Mathematics, vol. 50 (1980), Birkhauser · Zbl 0418.41012
[4] Cuyt, A., Handbook of Continued Fractions for Special Functions (2008), Springer Science
[5] Matala-aho, T., Type II Hermite- Padé approximations of generalized hypergeometric series, Constr. Approx., 33, 289-312 (2011) · Zbl 1236.41017
[6] Prévost, M., A new proof of the irrationality of \(\zeta(2)\) and \(\zeta(3)\) using Padé approximants, J. Comput. Appl. Math., 67, 2, 219-235 (1996) · Zbl 0855.11037
[7] Prévost, M., Remainder Padé approximants for the Hurwitz zeta function, Results Math., 74, 1, Article 51 pp. (2019), 22 · Zbl 1408.41008
[8] Prévost, M.; Rivoal, T., Application of Padé approximation to Euler’s constant and Stirling’s formula, Ramanujan J. (2020), in press
[9] Rivoal, T., Nombres d’Euler, approximants de Padé et constante de Catalan, Ramanujan J., 11, 2, 199-214 (2006) · Zbl 1152.11337
[10] Wilson, J., Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal., 11, 690-701 (1980) · Zbl 0454.33007
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