Special values of spectral zeta functions and combinatorics: Sturm-Liouville problems (Q6568847)

In this paper under review, the authors give a detailed review of the Sturm-Liouville problem and its spectral zeta functions \(\zeta_T(s)\). Then, they study the problem on the evaluation of special values \(\zeta_T(s)\) at positive integers \(s = n\) using the integral of Green functions. This method naturally involves an interesting combinatorial problem on counting permutations with fixed pinnacle and vale sets. As consequence, they derive a closed combinatorial formula for the Riemann zeta function and Bernoulli numbers at even integers \(\zeta(2n)\) and \(B_{2n}\), respectively.