On the differential transcendentality of the Morita \(p\)-adic gamma function
Differential Galois theory is a theory which is analogous to classical Galois theory in that it studies Galois groups of algebraic differential equations. The so-called \textit{Picard-Vessiot} theory developed the concept of differential Galois groups for differential field extensions, where \textit{differential field} means that it is equipped with a derivation. And, just like in the usual algebraic case, one can define differential algebraic or transcendental elements, depending on whether they solve an algebraic differential equation or not. This clearly written article consists of the proof that the \(p\)-adic-valued version of Euler's Gamma function, aka the Morita \(p\)-adic Gamma function, extended to a holomorphic function on \(\mathbb{C}_p\), the complex \(p\)-adic number field, does not satisfy any algebraic differential equation whose coefficients are rational functions with coefficients in \(\mathbb{C}_p\). In the language of differential Galois theory, this is stated as the Morita Gamma function being differentially transcendental over the function field \(\mathbb{C}_p(X)\). This is a \(p\)-adic analogue to Hölder's Theorem that Euler's Gamma function is differentially transcendental over \(\mathbb{C}(X)\). The proof is obtained by explicit calculations in the ring of differential polynomials over the complex \(p\)-adic function field of rational functions \(\mathbb{C}_p(X)\), and relies on the fact that the field \(\mathbb{C}_p\) is algebraically closed.
scientific article; zbMATH DE number 7659687