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Proof of conjectures on series with summands involving binom{2k}{k}8^k/(binom{3k}{k}binom{6k}{3k})
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Proof of conjectures on series with summands involving binom{2k}{k}8^k/(binom{3k}{k}binom{6k}{3k})
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Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\binom{2k}k8^k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that \[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,\pi+27\] and \[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{\pi^3}{12\sqrt2},\] where $\mathsf H^{(2)}_m$ denotes the second-order harmonic number $\sum_{0<j\leq m}\frac1{j^2}$.
Forward citations
Cited by 3 Pith papers
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Functional Dilogarithm Identities in Quadratic Fields
Ratio of sextic and cubic arctangent integrals equals a rational constant and yields new dilogarithm functional equations plus proofs of several prior conjectures.
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Functional Dilogarithm Identities in Quadratic Fields
Derives new 3- and 6-term dilogarithm functional equations and ladders from beta integrals, with analytic proofs of prior conjectures.
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Functional Dilogarithm Identities in Quadratic Fields
New functional dilogarithm identities and ladders are derived from integral/hypergeometric comparisons, with claimed analytic proofs of Loxton–Lewin and Bytsko identities.
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