Higher Apery-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator

TL;DR

This paper introduces and studies a generalization of Apery-like numbers related to the spectral zeta function of the non-commutative harmonic oscillator, revealing recurrence relations and congruence properties.

Contribution

It defines higher Apery-like numbers associated with the spectral zeta function and explores their recurrence relations and congruence properties, extending the classical Apery number theory.

Findings

Recurrence relations form a ladder structure among the numbers.

Congruence relations analogous to classical Apery numbers are established.

The rational parts of these numbers exhibit specific congruence patterns.

Abstract

A generalization of the Apery-like numbers, which is used to describe the special values $\zeta_Q(2)$ and $\zeta_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider the `rational part' of the higher Apery-like numbers. We discuss several kinds of congruence relations among them, which are regarded as an analogue of the ones among Apery numbers.