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The $$p$$-adic log gamma function and $$p$$-adic Euler constants. (English) Zbl 0382.12008
A locally analytic function $$G_p$$ on $$\mathbb C_p-\mathbb Z_p$$ is defined. $$G_p$$ satisfies relations similar to the standard formulas for log gamma (example: $$G_p(x+1)=G_p(x)+\log x$$, $$G_p(x)+G_p(1-x)=0$$, …), but $$G_p$$ is not the log of Morita’s gamma function $$\Gamma_p$$. $$p$$-adic Euler constants are defined and used to obtain a finite expression for $$G'_p$$ at rational points and for the logarithmic derivative of $$\Gamma_p$$ at certain rational points.

##### MSC:
 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 33B15 Gamma, beta and polygamma functions
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