On the Kimoto-Wakayama supercongruence conjecture on Apéry-like numbers (Q6641670)

The author proves that, letting \(p\) be an odd prime and \(\operatorname{ord}_{p} x\) be the exponent of \(p\) in \(x\), i.e., \(x=\prod_{p} p^{\operatorname{ord}_{p} x}\), \N\[\N\binom{2 j^{\prime}}{j^{\prime}} \equiv(-1)^{\frac{p-1}{2}}\binom{2 j}{j} \quad \pmod{p^{r+s+1}} \tag{1}\N\]\Nwhere \(2 j^{\prime}+1=(2 j+1) p\), \(r=\operatorname{ord}_{p}(2 j+1)\), and \(s=\operatorname{ord}_{p}\binom{2 j}{j}\).\N\NThe \((1)\) is an equivalent formulation of the following supercongruence \N\[\N p^{(2 s+1) n} \widetilde{J}_{2 s+1}\left(m p^{n}\right) \equiv p^{(2 s+1)(n-1)} \widetilde{J}_{2 s+1}\left(m p^{n-1}\right) \quad \pmod{p^{n}} \tag{2} \N\]\Nwhere \(m\) is a positive integer, with \(m \leq \frac{p-1}{2}\), and \N\[\N\widetilde{J}_{2 s+1}(n)=\sum_{k=0}^{n}(-1)^{k}\binom{-\frac{1}{2}}{k}^{2}\binom{n}{k} Z_{s}^{\text {odd }}(k) \tag{3}\N\]\NIn \((3)\) we have \( Z_{s}^{\text{odd}}(k) = \frac{(-1)^{s}}{2} \sum_{k>j_{1}>\cdots>j_{s} \geq 0} \frac{1}{\left(j_{1}+\frac{1}{2}\right)^{2} \cdots\left(j_{s-1}+\frac{1}{2}\right)^{2}\left(j_{s}+\frac{1}{2}\right)^{3}}\binom{-\frac{1}{2}}{j_{s}}^{-2}\).\N\NThe supercongruence \((2)\) for the odd case \((3)\) was conjectured by \textit {K. Kimoto} and \textit {M. Wakayama} [Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. 10, No. 2, 205--275 (2023; Zbl 1531.11083)] as corresponding of the even case \(\widetilde{J}_{2s+2}(n)\) they had established while studying the two-fold Apéry-like numbers \(\widetilde{J}_{k}(n)\) related to the special values of the spectral zeta function \(\zeta_{Q}(s)\), at integer points, associated to the non-commutative harmonic oscillator defined by the differential operator \((4)\) for \(\alpha, \beta>0\), with \(\alpha \beta>1\), \N\[\NQ_{\alpha, \beta}=\left(\begin{array}{cc} \alpha & 0 \\\N0 & \beta \end{array}\right)\left(-\frac{1}{2} \frac{d^{2}}{d x^{2}}+\frac{1}{2} x^{2}\right)+\left(\begin{array}{cc} 0 & -1 \\\N1 & 0 \end{array}\right)\left(x \frac{d}{d x}+\frac{1}{2}\right) \tag{4} \N\]\NBesides an interesting extension of Wolstenholme's theorem from \textit {C. Helou} and \textit {G. Terjanian} [J. Number Theory 128, No. 3, 475--499 (2008; Zbl 1236.11003)], the proof employs a binomial manipulation of \(\operatorname{ord}_{p}\binom{2j}{j}\) and an ingenious rearrangement of Morley's congruence.