On the series \(\sum ^{\infty}_{k=1}\binom{2k}{k}^{-1}k^{-n}\) and related sums. (English) Zbl 0551.10031
Motivated by certain series appearing in Apéry’s proof of the irrationality of \(\zeta(3)\) [Astérisque 61, 11–13 (1979; Zbl 0401.10049)] the author studies sums of the form
\[
s(n,x)=\sum^{\infty}_{k=1}\left( \begin{matrix} 2k\\ k\end{matrix} \right)^{-1}(2k)^{-n}x^{2k}
\]
and shows that \(s(n,1)\) is a \(\log \sin\) integral. The author evaluates these integrals for positive integer \(n\leq 5\) in terms of Dirichlet \(L\)-series with real characters. Connections are described with previous work of Ramanujan and D. Leshchiner [J. Number Theory 13, 355–362 (1981; Zbl 0468.10006)].
Reviewer: Tom M. Apostol (Pasadena)
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
References:
| [1] | Apèry, R., Irrationalitié de ζ(2) et ζ(3). Journees arithmétiques de Luminy, Astérisque, 61, 11-13 (1979) · Zbl 0401.10049 |
| [2] | B. C. Berndt and P. T. Joshi; B. C. Berndt and P. T. Joshi · Zbl 0519.40001 |
| [3] | Leschiner, D., Some new identities for \(ζ(k)\), J. Number Theory, 13, 355-362 (1981) · Zbl 0468.10006 |
| [4] | Lewin, L., (Dilogarithms and Associated Functions (1958), McDonald: McDonald London) · Zbl 0083.35904 |
| [5] | Ramanujan, S., (On Question 330 of Professor Sanjana, “Collected Papers” (1962), Chelsea: Chelsea New York), 15-17 |
| [6] | van der Poorten, A., A proof that Euler missed… Apèry’s proof of the irrationality of ζ(3), Math. Intelligencer, 1, 195-203 (1979) · Zbl 0409.10028 |
| [7] | van der Poorten, A., Some wonderful formulae… footnotes to Apèry’s proof of the irrationality of ζ(3), Sem. Delange-Pisot-Poitou, 20, 7 (1978-1979) |
| [8] | van der Poorten, A., Some wonderful formulas… an introduction to polylogarithm, (Ribenboim, P., Proc. Queen’s Number Theory Conf., 1979. Proc. Queen’s Number Theory Conf., 1979, Queen’s Papers in Pure and Applied Mathematics, No. 54 (1980)), 269-286, Kingston · Zbl 0448.10025 |
| [9] | Zucker, I. J.; Robertson, M. M., Some properties of Dirichlet \(L\)-series, J. Phys. A., 9, 1207-1214 (1976) · Zbl 0338.10037 |
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