Combinatorics of noncommutative harmonic oscillator determinant

Abstract

We obtain an explicit evaluation of the determinants that appear in the denominators of the integrands for the iterated integrals expressing special values of the spectral zeta function of the noncommutative harmonic oscillator at integers greater than 1. The determinants in question are those of cyclic tridiagonal matrices; their explicit evaluation is carried out, in addition to the inclusion-exclusion principle, conceptionally using combinatorial arguments based on monomer-dimer systems. This explicit evaluation serves as a core component in the study of arithmetic properties such as the modularity of the special values of the spectral zeta function.