Ljunggren--Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients
Motivated by the polynomial form of Kalinin's Gaussian analogue of Wolstenholme's theorem, following Kalinin, we study a two-dimensional factorial ratio over the Gaussian integers, which we call the rectangular Gaussian binomial coefficient. For a rational prime $p\equiv 3\pmod 4$, we first prove that these coefficients are $p$-integral. Our main result is a Ljunggren--Jacobsthal-type supercongruence: for $p>5$ and every $k\geq 1$, simultaneous dilation of all four parameters by $p^k$ changes the coefficient by a multiple of $p^{3k}$. In particular, this proves the inert-prime case of a conjecture of Kalinin. We also establish a rectangular Bailey-type congruence modulo $p$ for parameters consisting of a large $p$-multiple and one base-$p$ digit. Its shape parallels Bailey's prime-power refinements of Lucas's theorem, but two additional ordinary binomial factors occur, reflecting the vertical and horizontal boundary strips of a rectangular block decomposition. The proofs combine reciprocal-power-sum estimates in $\mathbb{Z}_p[i]$ with factorizations of rectangular products into complete $p^k\times p^k$ blocks.