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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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