## 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

### Keywords:

p-adic gamma function; locally analytic function

### Citations:

Zbl 0542.00006; Zbl 0532.12017