A study on algebraic differential equations of Gamma function and Dirichlet series

The paper concerns the problem of algebraic differential independence between the gamma function and the function in a certain class F, which contains Dirichlet L-functions, L-functions in the extended Selberg class and some periodic functions. It is proved that the gamma function and the function in F cannot satisfy a class of algebraic differential equations with meromorphic coefficients ϕ having Nevanlinna characteristic satisfying T(r,ϕ)=o(r) as r→∞.