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

We derive an expression for the value zeta(Q)(3) of the spectral zeta function zeta(Q)(s) for the non-commutative harmonic oscillator using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted (J) over tilde (2)(n) and (J) over tilde (3)(n), which can be regarded as analogues of the Apery numbers, naturally arise and play a key role in obtaining the expressions for the values zeta(Q)(2) and zeta(Q)(3). We also show that the numbers (J) over tilde (2)(n) and (J) over tilde (3)(n) have congruence relations such as those of the Apery numbers.