Diamond, Jack The \(p\)-adic log gamma function and \(p\)-adic Euler constants. (English) Zbl 0382.12008 Trans. Am. Math. Soc. 233, 321-337 (1977). 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. Reviewer: Gilles Christol (Paris) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 4 ReviewsCited in 39 Documents MSC: 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) 33B15 Gamma, beta and polygamma functions PDF BibTeX XML Cite \textit{J. Diamond}, Trans. Am. Math. Soc. 233, 321--337 (1977; Zbl 0382.12008) Full Text: DOI OpenURL