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

Zhi-Wei Sun; Yajun Zhou

paper

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

arXiv:2401.14197

Abstract

Using cyclotomic multiple zeta values of level $8$ , we confirm and generalize several conjectural identities on infinite series with summands involving $(k 2 k) 8^{k} / ((k 3 k) (3 k 6 k))$ . For example, we prove that \[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,π+27\] and \[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{π^3}{12\sqrt2},\] where $H_{m}^{(2)}$ denotes the second-order harmonic number $\sum_{0 < j \leq m} \frac{1}{j ^{2}}$ .

23 pages, 5 tables. A sequel to arXiv:2401.12083v1

Proof of conjectures on series with summands involving (k2k​)8k/((k3k​)(3k6k​))

Abstract

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