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New \(_{5}F_{4}\) hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/ \pi \). (English) Zbl 1226.11113

In this paper the author obtains some formulas for three variable Mahler measures in terms of \(_5F_4\) hypergeometric function. Using this he derives some new formulas for \(1/\pi\). For example, if \(a_n=\sum_{k=0}^n {2n-2k \choose n-k} {2k \choose k} {n \choose k}^2\) then \[ \frac{2}{\pi}= \sum_{n=0}^{\infty} (-1)^n \frac{(3n+1)a_n}{32^n} \] and \[ \frac{8\sqrt{3}}{3\pi}= \sum_{n=0}^{\infty} \frac{(5n+1)a_n}{64^n}. \]

MSC:

11R09 Polynomials (irreducibility, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C05 Classical hypergeometric functions, \({}_2F_1\)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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References:

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