Evaluations of ∑_k=1^∞ \fracx^kk²\binom3kk and related series

2024/01/22 by Sun, Zhi-Wei, Zhou, Yajun
#05A10 #11B65 #11M32 #33B15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2401.12083

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 ∑_k=0^∞\fracx₀^k(k+1)\binom3kk and ∑_k=1^∞\fracx₀^kk²\binom3kk for any x₀∈(-27/4,27/4).

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