Proof of a conjecture due to Sun concerning Catalan's constant

John Campbell (1)

1 Department of Mathematics and Statistics [Toronto]

In 2014, Sun had conjectured that the following formula involving Catalan's constant $G$ holds, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ denote the $n^{\text{th}}$ harmonic number: $$ \sum_{k=0}^{\infty} \frac{\binom{2k}{k}}{(2k+1)16^k} \left( 3 H_{2k+1} + \frac{4}{2k+1} \right) = 8 G. $$ This had been discovered experimentally, in the context of the study of special values of $L$-functions. In this article, we prove this conjecture, using various identities concerning the dilogarithm function, and using an argument relying on a ${}_{3}F_{2}\left( \frac{1}{4} \right)$-series for Gieseking's constant. It appears that this conjecture due to Sun has not previously been solved, based on the relevant literature.

Mots clés: Catalan's constant; Harmonic number; Dilogarithm; L-function

Domaines: Mathématiques [math]

John Campbell. Proof of a conjecture due to Sun concerning Catalan's constant. 2022. ⟨hal-03644515⟩