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 oscillators (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations. We study first the 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 Apery-like numbers defined through the analysis of special values $\zeta_Q(n)$. Particularly, we show the generating function $w_{2n}$ of such Apery-like numbers appearing (as the ``first anomaly'') in $\zeta_Q(2n)$ for $n=2$ is given by an \emph{automorphic integral with rational period functions} in the sense of Knopp, but still remains to be clarifying 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 $1$. 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 cohomology groups.