Explicit evaluation of the derivatives of $p$-adic Gamma function at 0

Question

The definition of the $p$-adic Gamma function $\Gamma_p(x)$ for an odd prime number $p$ can be found in the book "A Course in $p$-adic analysis" by A. M. Robert. While the construction of $\log \Gamma_p(x)$ is also included in the book. The function $\log \Gamma_p(x)$ is odd, and has a series expansion in the variable $x$. For more details, see Chapter 7 of the book. \begin{equation} \log \Gamma_p(x)=(\log \Gamma_p)^{(1)}(0)x+\frac{1}{3!}(\log \Gamma_p)^{(3)}(0)x^3+\frac{1}{5!}(\log \Gamma_p)^{(5)}(0)x^5+\cdots. \end{equation} Since $\log \Gamma_p(x)$ is an odd function, we have \begin{equation} (\log \Gamma_p)^{(2n)}(0)=0. \end{equation} I am wondering whether there is an explicit method to evaluate the coefficient $(\log \Gamma_p)^{(s)}(0)$, at least the first several terms $(\log \Gamma_p)^{(3)}(0)$, $(\log \Gamma_p)^{(5)}(0)$, and $(\log \Gamma_p)^{(7)}(0)$?

One helpful observations is that since we have \begin{equation} \Gamma_p(0)=1, \end{equation} so $(\log \Gamma_p)^{(s)}(0)$ can be computed by \begin{equation} \Gamma_p^{'}(0),\Gamma^{''}_p(0),\Gamma^{'''}_p(0), \cdots,\Gamma^{(s)}_p(0). \end{equation} So this question is equivalent to evaluate $\Gamma_p^{'}(0),\Gamma^{''}_p(0),\Gamma^{'''}_p(0),\cdots$. Does anyone know an explicit formula to compute them?

Answer 1

This is very classical, and the formula is very similar to the complex case: the coefficients of the Taylor expansion of $\log(\Gamma_p(x))$ are essentially $p$-adic Bernoulli numbers, or equivalently values at integers of the Kubota--Leopoldt $p$-adic zeta function. This is done in complete detail in particular in my book GTM 240, Chapter 11, specifically Propositions 11.3.15 and 11.5.19, Section 11.6.5, and Exercise 49 of Chapter 11.