On a result of Koecher concerning Markov-Ap\'ery type formulas for the Riemann zeta function

TL;DR

This paper extends Koecher's method to derive new identities for the Riemann zeta function, Euler sums, and powers of pi, broadening the understanding of these special mathematical constants.

Contribution

It generalizes Koecher's approach, producing new identities for zeta values, Euler sums, and powers of pi, with applications to classical and modern problems.

Findings

Derived new identities for zeta(3) and other odd integers.

Established infinite classes of identities for alternating Euler sums.

Produced identities for even powers of pi.

Abstract

Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for $\zeta(3)$ due to Markov and rediscovered by Ap\'ery. In this paper we extend Koecher's method to a very general setting and prove two more specific but still rather general results. As applications we obtain infinite classes of identities for alternating Euler sums, further Markov-Ap\'ery type identities, and identities for even powers of $\pi$