A new series for ${\pi }^3$ and related congruences

Abstract

Let $H_n^{(2)}$ denote the second-order harmonic number ${\sum }_{0<k\le n}1/k^2$ for $n=0,1,2,\dots$. In this paper we obtain the following identity: $$\sum _{k=1}^{\infty }\frac{2^k{H}_{k-1}^{(2)}}{k\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}=\frac{{\pi }^3}{48}.$$ We explain how we found the series and develop related congruences involving Bernoulli or Euler numbers; for example, it is shown that $$\sum _{k=1}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{2^k}{H}_k^{(2)}\equiv -E_{p-3}(\mathrm{mod}p)$$ for any prime $p>3$, where $E_0,E_1,E_2,\dots$ are Euler numbers. Motivated by the Amdeberhan-Zeilberger identity ${\sum }_{k=1}^{\infty }(21k-8)/(k^3{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3)={\pi }^2/6$, we also establish the congruence $$\sum _{k=1}^{(p-1)/2}\frac{21k-8}{k^3{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3}\equiv (-1{)}^{(p+1)/2}4E_{p-3}(\mathrm{mod}p)$$ for each prime $p>3$.

Final published version

References in corpus (3)

• On congruences related to central binomial coefficients
• On sums of binomial coefficients modulo p^2
• Supercongruences for a truncated hypergeometric series

Cited by in corpus (3)

• Open Conjectures on Congruences
• Discriminants in the Grothendieck Ring
• Congruences arising from Apéry-type series for zeta values