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Evaluations of $ \sum_{k=1}^\infty \frac{x^k}{k^2\binom{3k}{k}}$ and related series
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abstract
We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$.
Forward citations
Cited by 2 Pith papers
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Derivative Sums of Balanced Gamma Quotients and Multiple Zeta Values: Five Conjectures of Zhi-Wei Sun
Balanced gamma quotients reduce derivative sums to ordinary multiple zeta values, proving and correcting four conjectures of Sun without numerical fitting.
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Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations
Weighted master identity for Γ(x)²/(2Γ(2x)) gives all-derivative-order sums in zeta and log-sine values and proves Sun's Conjecture 4.1.
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