On a p-adic interpolating power series of the generalized Euler numbers (Q919034)

Let p be a prime number and \(\chi\) a primitive Dirichlet character with conductor \(f_{\chi}\). Then for an algebraic number \(u\neq 1\) the n-th generalized Euler number \(H^ n_{\chi}(u)\) defined by \[ \sum^{f_{\chi}-1}_{a=0}\frac{(1-u^{f_{\chi}})\chi (a)e^{at}u^{f_{\chi}-a-1}}{e^{f_{\chi}t}- u^{f_{\chi}}}=\sum^{\infty}_{n=0}H^ n_{\chi}(u)t^ n/(n!). \] By similar methods used in \textit{W. Sinnott}'s papers [Invent. Math. 75, 273-282 (1984; Zbl 0531.12004); J. Reine Angew. Math. 382, 22-34 (1987; Zbl 0621.12015); Invent. Math. 89, 139-157 (1987; Zbl 0637.12004)] the author constructs the corresponding p-adic interpolating function \(\ell_ p(s,u,\chi)\) with the help of the interpolating power series \(F_{\chi,u}(T)\) and calculates the \(\mu\)-invariant of \(F_{\chi,u}(T)\) [see the author, Mem. Fac. Sci., Kyushu Univ., Ser. A 43, 43-53 (1989; see the following review)]. The analytic properties of \(F_{\chi,u}(T)\) and the p-adic valuation of the generalized Euler numbers are considered.