Special values of spectral zeta functions and combinatorics: Sturm–Liouville problems.

In this paper, we apply the combinatorial results on counting permutations with fixed pinnacle and vale sets to evaluate the special values of the spectral zeta functions of Sturm–Liouville differential operators. As applications, we get a combinatorial formula for the special values of spectral zeta functions and give a new explicit formula for Bernoulli numbers. [ABSTRACT FROM AUTHOR]