Central binomial sums, multiple Clausen values, and zeta values. (English) Zbl 0998.11045

Borwein, Jonathan Michael; Broadhurst, David J.; Kamnitzer, Joel

Exp. Math. 10, No. 1, 25-34 (2001).

The goal of this paper is the evaluation (in terms of more basic constants) of the central binomial sums \(S(k):=\sum_{n=1}^\infty \frac{1}{n^k {2n \choose n}}\) for integers \(k\). Classical evaluations are \(S(2)=\frac 13 \zeta(2)\) and \(S(4)=\frac{17}{36} \zeta(4)\). The authors’ new evaluations are based on a connection (via log-sine integrals) of the central binomial sums to multiple Clausen values (MCV’s) \(\mu(a_1,\dots,a_k):=\text{Li}_{a_1,\dots,a_k}(\omega)\), where \(\omega=e^{i\pi/3}\) is the sixth root of unity, and \(\text{Li}_{a_1,\dots,a_k}(z):=\sum_{n_1>\dots>n_k>0} \frac{z^{n_1}}{n_1^{a_1}\cdots n_k^{a_k}}\) is a multidimensional polylogarithm. Examples of such new evaluations are \(S(3)=\frac 23 \pi \operatorname{Im}(\mu(2))-\frac 43 \zeta(3)\) or \(S(6)=-\frac 43 \pi \operatorname{Im}(\mu(4,1))+\frac{3341}{1296} \zeta(6)-\frac 43 \zeta(3)^2\). The authors then investigate relationships (duality formulas) between sets of different MCV’s, and they use these and generating function results to give explicit evaluations for certain classes of MCV’s. Finally, they give some evaluations for the alternating binomial series (Apéry sums) \(A(k):=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^k {2n \choose n}}\).

Reviewer: Roland Girgensohn (Neuherberg)

MSC:

11M06	\(\zeta (s)\) and \(L(s, \chi)\)
05A10	Factorials, binomial coefficients, combinatorial functions
11B75	Other combinatorial number theory
33B30	Higher logarithm functions

Keywords:

binomial sums; multiple zeta values; log-sine integrals; Clausen’s function; multiple Clausen values; polylogarithms; Apéry sums

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