A few remarks on linear forms involving Catalan's constant

Abstract

In the joint work of T.Rivoal and the author, a hypergeometric construction was proposed for studing arithmetic properties of the values of Dirichlet's beta function $β(s)$ at even positive integers. The construction gives some bonuses for Catalan's constant $G=β(2)$, such as a second-order Apery-like recursion and a permutation group in the sense of G.Rhin and C.Viola. Here we prove expected integrality properties of solutions to the above recursion as well as suggest a simpler (also second-order and Apery-like) one for $G$. We "enlarge" the permutation group of our previous (joint with T.Rivoal) work by showing that the total 120-permutation group for $ζ(2)$ (known thanks to G.Rhin and C.Viola) can be applied in arithmetic study of Catalan's constant. These considerations have computational meanings and do not allow us to prove the (presumed) irrationality of $G$. Finally, we suggest a conjecture yielding the irrationality property of numbers (e.g., of Catalan's constant) from existence of suitable second-order difference equations (recursions).