Rapidly converging series for \(\zeta(2 n + 1)\) from Fourier series
Summary: Ever since Euler first evaluated \(\zeta(2)\) and \(\zeta(2 m)\), numerous interesting solutions of the problem of evaluating the \(\zeta(2 m)\) (\(m \in \mathbb N\)) have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of \(\zeta(2 m)\) (\(m \in \mathbb N\)) is known for \(\zeta(2 m + 1)\) (\(m \in \mathbb N\)) or even for any special case such as \(\zeta(3)\). Instead, various rapidly converging series for \(\zeta(2 m + 1)\) have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for \(\zeta(2 m + 1)\). In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of \(\beta(2 m + 1)\) and \(\zeta(2 m)\) (\(m \in \mathbb N\)), which have been treated in earlier works.
scientific article; zbMATH DE number 7022416
scientific article