The aim of this article is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of differential fields and properties of certain known transcendental differential functions, as of $\Gamma(x)$. Furthermore, it also determines differential transcendence of solution of some functional equations.
"Differentially transcendental functions." Bull. Belg. Math. Soc. Simon Stevin 15 (2) 193 - 201, May 2008. https://doi.org/10.36045/bbms/1210254818