A new series for $π^3$ and related congruences

Details A new series for $π^3$ and related congruences A new series for $π^3$ and related congruences

2010-09-27

arxiv.org/abs/1009.5375v7

Mathematics

Number Theory

25 pages, polished version

Scientific paper

Let $H_n^{(2)}$ denote the second-order harmonic number $\sum_{03, where $E_0,E_1,E_2,...$ are Euler numbers. Motivated by the Amdeberhan-Zeilberger identity $\sum_{k>0}(21k-8)/(k^3\binom{2k}{k}^3)=\pi^2/6$, we also establish the congruence $$\sum_{k=1}^{(p-1)/2}(21k-8)/(k^3*\binom{2k}{k}^3)=(-1)^{(p+1)/2}4E_{p-3} (mod p)$$ for each prime $p>3$.

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