We derive an expression for the value ζQ(3) of the spectral zeta function ζQ(s) for the non-commutative harmonic oscillator using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted J̃2(n) and J̃3(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 ζQ(2) and ζ Q(3). We also show that the numbers J̃ 2(n) and J̃3(n) have congruence relations such as those of the Apéry numbers.
All Science Journal Classification (ASJC) codes
- 数学 (全般)