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Uniqueness of \(\Gamma _ p:\) The locally analytic case. (English) Zbl 0552.12009

Cohomologie p-adique, Astérisque 119/120, 9-15 (1984).
[For the entire collection see Zbl 0542.00006.]
Let \(\Omega\) be a p-adic universal domain and for \(\rho >0\), let \(W_{\rho}({\mathbb{Z}})=\{x\in \Omega | | x-z| <\rho\), for some \(z\in {\mathbb{Z}}\}\). For a p-adic integer \(x=\sum_{i=0}x_ i p^ i,\quad 0\leq x_ i<p,\) we put \(\phi (x)=\sum_{i=1}x_ i p^{i-1}\). The author proves the following theorem: Fix \(\rho\leq 1\) and let \(F: W_ p({\mathbb{Z}})\to \Omega\) be a non-vanishing, locally analytic function satisfying for all positive integers n the property: \[ (P)\quad if\quad a\in {\mathbb{Z}}_ p,\quad \phi^ n(a)=a,\quad then\quad \prod^{n- 1}_{i=0}F(\phi^ ia)=1. \] Then there exists a non-vanishing, locally analytic function \(G: W_{p\rho}({\mathbb{Z}})\to \Omega\) such that for all \(x\in W_{\rho}({\mathbb{Z}})\), \(F(x)=G(x)/G(\phi x).\)- Functions satisfying property (P) appear in the p-adic theory of the gamma function [cf. the author, Trans. Am. Math. Soc. 278, 57-63 (1983; Zbl 0532.12017)].
Reviewer: F.Baldassarri

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14F30 \(p\)-adic cohomology, crystalline cohomology
33B15 Gamma, beta and polygamma functions