A class of series representations for Catalan’s constant

Abstract

Let

$$ S(j)= \sum _{\nu =1}^\infty \frac{\nu }{16^\nu (2\nu -1)^2 (2\nu +1)(2\nu +j)}{2\nu \atopwithdelims ()\nu }^2, \quad j\in \{2,3,4,... \}. $$

In 2022, N. Bhandari showed that for \(j\in \{3,4,5\}\) there are rational numbers \(a_j\) and \(b_j\) such that

$$ 4 \pi S(j)=a_j-b_ jG, $$

where G denotes the Catalan constant. He conjectured that this representation holds for all \(j\ge 3\). Here, we prove this conjecture. More precisely, we offer recursion formulas to determine the numbers \(a_j\) and bj \((j\ge 3)\) explicitly.