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

Abstract

We derive an expression for the value ${\zeta }_Q(3)$ of the spectral zeta function ${\zeta }_Q(s)$ 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 ${\zeta }_Q(2)$ and ${\zeta }_Q(3)$. We also show that the numbers $J_2(n)$ and $J_3(n)$ have congruence relations like those of the Apéry numbers.

arXiv comment18 pages

References in corpus (1)

• A special value of the spectral zeta function of the non-commutative harmonic osciallators

Cited by in corpus (1)

• Special value formula for the spectral zeta function of the non-commutative harmonic oscillator