Generalized Hölder's theorem for Vignéras' multiple gamma function (Q5946912)

The multiple gamma function is defined by NEWLINE\[NEWLINEG_r(z+1)= G_{r-1}(z) G_r(z),\;G_r(1)=1,\;{d^{r+1} \over dz^{r+1}} \log G_r(z+1)\geq 0NEWLINE\]NEWLINE for \(z\geq 0,\;G_0(z)=z.\) The gamma function is therefore \(\Gamma(z)= G_1(z)\). Hölder proved that the gamma function does not satisfy any algebraic differential equation over the rational field \(\mathbb{C}(z)\). In the paper under review, it is shown that the multiple gamma function does not satisfy any such differential equation either. A new method for determining the factors in the Weierstrass infinite product representation for the multiple gamma function is also given.