On the Kimoto–Wakayama supercongruence conjecture on Apéry-like numbers

Kimoto and Wakayama [Ann. Inst. Henri Poincaré D 10 (2023), 205–275] studied the special values of the spectral zeta function \(\zeta _Q(s)\) ${\zeta }_Q(s)$ associated to the non-commutative harmonic oscillator \(Q_{\alpha ,\beta }\) $Q_{\alpha ,\beta }$ . Two kinds of Apéry-like numbers (even case \(\widetilde{J}_{2s+2}(n)\) ${\tilde{J}}_{2s+2}(n)$ and odd case \(\widetilde{J}_{2s+1}(n)\) ${\tilde{J}}_{2s+1}(n)$ ) naturally arise in the expressions for the special values of \(\zeta _Q(s)\) ${\zeta }_Q(s)$ at integer points. Supercongruences among these Apéry-like numbers lead one to the modularity of the generating functions of the Apéry-like numbers. Kimoto and Wakayama established a supercongruence among \(\widetilde{J}_{2s+2}(n)\) ${\tilde{J}}_{2s+2}(n)$ , and conjectured the same type of supercongruence for \(\widetilde{J}_{2s+1}(n)\) ${\tilde{J}}_{2s+1}(n)$ as in the even case \(\widetilde{J}_{2s+2}(n)\) ${\tilde{J}}_{2s+2}(n)$ . In this work, we confirm Kimoto and Wakayama’s supercongruence conjecture in the odd case of Apéry-like numbers \(\widetilde{J}_{2s+1}(n)\) ${\tilde{J}}_{2s+1}(n)$ .