Ap\'ery-like numbers for non-commutative harmonic oscillators and automorphic integrals

The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function $\zeta_Q(s)$ of the NcHO at $s=n$ $(n=2,3,\dots)$ and then the generating and meta-generating functions for Ap\'ery-like numbers defined through the analysis of special values $\zeta_Q(n)$. Actually, we show that the generating function $w_{2n}$ of such Ap\'ery-like numbers appearing (as the "first anomaly") in $\zeta_Q(2n)$ for $n=2$ gives an example of automorphic integral with rational period functions in the sense of Knopp, but still a better explanation remains to be clarified explicitly for $n>2$. This is a generalization of our earlier result on showing that $w_2$ is interpreted as a $\Gamma(2)$-modular form of weight $1$. Moreover, certain congruence relations over primes for "normalized" Ap\'ery-like numbers are also proven. In order to describe $w_{2n}$ in a similar manner as $w_2$, we introduce a differential Eisenstein series by using analytic continuation of a classical generalized Eisenstein series due to Berndt. The differential Eisenstein series is actually a typical example of the automorphic integral of negative weight. We then have an explicit expression of $w_4$ in terms of the differential Eisenstein series. We discuss also shortly the Hecke operators acting on such automorphic integrals and relating Eichler's cohomology group.