Some fast convergent series for the mathematical constants $\zeta(4)$ and $\zeta(5)$

TL;DR

This paper proves four conjectural series for the constants (4) and (5) proposed by Sun, using operator methods and hypergeometric transformations, and introduces new series for these constants.

Contribution

The paper provides rigorous proofs for Sun's conjectural series for (4) and (5), and discovers new series representations for these constants.

Findings

Proved four conjectural series for (4) and (5).

Introduced new series for (4) and (5).

Applied operator methods and hypergeometric transformations.

Abstract

Recently, Sun [preprint, arXiv: 2210.07238v7] proposed two conjectural series for the mathematical constant $\zeta(4)$ and two conjectural series for the mathematical constant $\zeta(5)$. In terms of the operator method and two hypergeometric transformations, we prove these four conjectures. Furthermore, we also find some new series for the two constants in this paper.