The q-analogue of the p-adic gamma function
Let p be an odd prime, and let \({\bar {\mathbb{C}}}_ p\) be the completion of the algebraic closure of the p-adic field \({\mathbb{Q}}_ p\). For \(x\in {\mathbb{C}}_ p\), let \(| x|_ p\) be the p-adic absolute value, normalized so that \(| p|_ p=p^{-1}\). Let \(q\in {\mathbb{C}}_ p\) satisfy \(| q-1|_ p<| p|_ p^{1/(p-1)}\). The author defines p-adic functions \(G_{p,q}\) and \(G^*_{p,q}\), depending on the parameter q, which are generalizations of Diamond's p-adic log-gamma functions \(G_ p\) and \(G^*_ p\), analogous to Koblitz' generalization \(\Gamma_{p,q}\) of Morita's gamma function \(\Gamma_ p\) (although Koblitz defined his function for any \(| q-1| <1)\). Koblitz previously defined functions \(\psi_{p,q}\) and \(\psi^*_{p,q}\) for \(| q-1| <1\), which, in the cases where both are defined, are the derivatives of \(G_{p,q}\) and \(G^*_{p,q}.\) The author's functions are locally analytic on \(B-{\mathbb{Z}}_ p\) and \(B- {\mathbb{Z}}^*_ p\) respectively, where B is the open ball in \({\mathbb{C}}_ p\) of radius \(| p|_ p^{1/(p-1)}/| 1-q|_ p\). He proves the appropriate generalization of the theorems of Diamond and Ferrero-Greenberg relating \(\log (\Gamma_ p)\) to \(G_ p\) and \(G^*_ p\). He also proves a formula relating \(G_{p,q}\) and \(G^*_{p,q}\), and proves a difference equation, multiplication theorem, and reflection formula for them. These generalize theorems proved by Koblitz for \(\psi_{p,q}\) and \(\psi^*_{p,q}\). The key ingredient in the construction of \(G_{p,q}\) and \(G^*_{p,q}\) is the p-adic dilogarithm.
scientific article; zbMATH DE number 4085774