A class of series acceleration formulae for Catalan's constant

TL;DR

This paper develops new transformation and series acceleration formulas for Catalan's constant using log tangent integrals, enabling more efficient computation and closed-form expressions involving fundamental constants.

Contribution

It introduces novel series acceleration formulae for Catalan's constant derived from transformation formulas for the log tangent integral, with explicit closed-form representations.

Findings

Series converge faster for Catalan's constant

Explicit closed-form expressions involving Catalan's constant and logarithms

New transformation formulas for the log tangent integral

Abstract

In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae are characterized by the presence of an infinite series whose general term consists of a linear recurrence damped by the central binomial coefficient and a certain quadratic polynomial. Typically, the series can be expressed in closed form as a rational linear combination of Catalan's constant and pi times the logarithm of an algebraic unit.