In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor H. Moll.
It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
Definition
The generalized polygamma function is defined as follows:
: \psi(z,q)=\frac{\zeta'(z+1,q)+(\psi(-z)+\gamma ) \zeta (z+1,q)}{\Gamma (-z)} \,
or alternatively,
: \psi(z,q)=e^{- \gamma z}\frac{\partial}{\partial z}\left(e^{\gamma z}\frac{\zeta(z+1,q)}{\Gamma(-z)}\right),
where \psi(z) is ...
Let $H$ be a subgroup of a group $G$ and, for $x\in G$, let $x^{-1}ax$ denote the set $\{x^{−1}ax : a\in H\}$. Prove that $x^{−1}Hx$ is a subgroup of $G$.
