APÉRY-LIKE NUMBERS ARISING FROM SPECIAL VALUES OF SPECTRAL ZETA FUNCTIONS FOR NON-COMMUTATIVE HARMONIC OSCILLATORS

Abstract

We derive an expression for the value ζ_Q(3) of the spectral zeta function ζ_Q(s) for the non-commutative harmonic oscillator using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted ${\\ ilde J}\\sb 2 (n)$ and ${\\ ilde J}\\sb 3 (n)$, 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 ${\\ ilde J}\\sb 2 (n)$ and ${\\ ilde J}\\sb 3 (n)$ have congruence relations such as those of the Apéry numbers.