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Proof of conjectures on series with summands involving $ \binom{2k}{k}8^k/(\binom{3k}{k}\binom{6k}{3k})$ (2401.14197v1)

Published 25 Jan 2024 in math.CA and math.NT

Abstract: Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\binom{2k}k8k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that [\sum_{k=0}\infty\frac{(350k-17)\binom{2k}k8k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,\pi+27] and [\sum_{k=1}\infty\frac{\left{(5k-1)\left[16\mathsf H_{2k-1}{(2)}-3\mathsf H_{k-1}{(2)}\right]-\frac{12(6k-1)}{(2k-1)2}\right}\binom{2k}k8k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{\pi3}{12\sqrt2},] where $\mathsf H{(2)}_m$ denotes the second-order harmonic number $\sum_{0<j\leq m}\frac1{j2}$.

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