Paper detail

Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators

We derive an expression for the value $ζ_Q(3)$ of the spectral zeta function $ζ_Q(s)$ studied by Ichinose and Wakayama for the non-commutative harmonic oscillator defined in the work of Parmeggiani and Wakayama using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted $J_2(n)$ and $J_3(n)$, which can be regarded as analogues of the Apéry numbers, naturally arise and play a key role in obtaining the expressions for the values $ζ_Q(2)$ and $ζ_Q(3)$. We also show that the numbers $J_2(n)$ and $J_3(n)$ have congruence relations like those of the Apéry numbers.