The q-analogue of Hölder's theorem for the gamma function
The basic gamma function \(\Gamma_ q(x)\) satisfies the recurrence relation \(\Gamma_ q(x+1)(1-q)=\Gamma_ q(x)(1-q^ x).\) By considering the logarithmic derivative of the basic gamma function g(x) which satisfies the recurrence relation \(g(x+1)=g(x)(q^ x\ln q)/(1-q^ x),\) it is shown that g(x) cannot satisfy any algebraic differential equation of finite order. This same result also applies to the basic gamma function itself which establishes the required basic analogue of Hölder's theorem.
scientific article; zbMATH DE number 3867724