Integrality conditions of certain hypergeometric coefficients: Generalization of a theorem of Landau (Q1343265)

A classic result of Landau states that given a finite family of linear forms \(u\) on \(\mathbb{R}^n\) with coefficients in \(\mathbb{Z}\), the product of factorials \(\prod_u u(X) !^{\varepsilon_u}\), \(\varepsilon_u = \pm 1\), is an integer for each \(X\in \mathbb{Z}^n\) for which \(u(X)\) is a positive integer if and only if \(\sum_u \varepsilon_u \lfloor u(X)\rfloor \geq 0\) for each \(X\in \mathbb{R}^n\) for which \(u(X)\) is positive. The author extends this result, finding necessary and sufficient conditions for the integrality of \(\prod_u u(X) !^{\varepsilon_u}\) where the \(u\) are homogeneous functions of the same degree. As a corollary, the author deduces that if \[ {kx!ky! \over \sqrt {x^2+y^2} !(k-1)x! (k-1)y!} \] is an integer, then it is divisible by \[ \prod_{j\geq 1} \left( {x+y- \sqrt {x^2+y^2} \over 2^j} \right)!. \]