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The Apéry numbers, the Almkvist-Zudilin numbers and new series for \(1/\pi\). (English) Zbl 1193.11038

Let \(\alpha_n\) denote the Apéry number \[ \alpha_n := \sum_{j=0}^n \binom{n}{j}^2\binom{n+j}{j}^2. \] In (apparently) unpublished work, T. Sato gave the following series for \(1/\pi\): \[ \frac{1}{\pi} = (72\sqrt{15} - 160\sqrt{3})\sum_{n \geq 0} \alpha_n \left(\frac{1}{2} - \frac{3\sqrt{5}}{20} + n \right) \left(\frac{1-\sqrt{5}}{2}\right)^{12n}. \] Later H. H. Chan, S. H. Chan and Z. Liu [Adv. Math. 186, No. 2, 396–410 (2004; Zbl 1122.11087)] gave a similar formula in terms of the “Domb numbers” \[ \tilde{\beta}_n := \sum_{j =0}^n \binom{n}{j}^2\binom{2j}{j}\binom{2(n-j)}{n-j}, \] namely \[ \frac{8}{\sqrt{3}\pi} = \sum_{n \geq 0}\tilde{\beta}_n(1+5n)\left(\frac{1}{64}\right)^n. \] In this paper the authors prove the two formulas \[ \frac{3\sqrt{3}}{2\pi} = \sum_{n \geq 0} \gamma_n(4n+1)\left(\frac{1}{81}\right)^n \] and \[ \frac{3\sqrt{3}}{\pi} = \sum_{n \geq 0} \gamma_n(4n+1)\left(\frac{-1}{27}\right)^n, \] where \(\gamma_n\) is the “Almkvist-Zudilin number”, \[ \gamma_n := \sum_{j=0}^n (-1)^{n-j}\frac{3^{n-3j}(3j)!}{(j!)^3}\binom{n}{3j}\binom{n+j}{j}. \]
There are similarities between the present methods and those of previous studies, but here the authors argue in greater generality which allows them to empirically discover many more formulas like those above. Twenty-one such formulas are presented (generally without proof), nine involving \(\tilde{\beta}_n\) and six involving each of \(\alpha_n\) and \(\gamma_n\).

MSC:

11F11 Holomorphic modular forms of integral weight
11F20 Dedekind eta function, Dedekind sums
11Y60 Evaluation of number-theoretic constants
11F03 Modular and automorphic functions
05A10 Factorials, binomial coefficients, combinatorial functions

Citations:

Zbl 1122.11087

Software:

PARI/GP
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