On the behavior of some two-variable \(p\)-adic \(L\)-functions.
The Kubota--Leopoldt \(p\)-adic \(L\)-function \(L_p(s,\chi)\) possesses a natural extension to a two variable function \(L_p(s,t,\chi)\) introduced by \textit{G.\ Fox} [Enseign. Math., II. Sér. 46, No. 3--4, 225--278 (2000; Zbl 0999.11073)]. This function is defined for all \(t\) in the valuation ring of \({\mathbb C}_p\) and satisfies \(L_p(s,0,\chi)=L_p(s,\chi)\). The author expresses \(L_p(s,t,\chi)\) by means of a \(p\)-adic integral. For \(\chi\) a power of the Teichmüller character, this representation allows him to extend the domain of definition of \(L_p(s,t,\chi)\) to include \(t\in q^{-1}{\mathbb Z}_p\) (here, as usual, \(q\) equals \(p\) or 4 according as \(p>2\) or \(p=2\), respectively). As another consequence the author obtains new systems of congruences for generalized Bernoulli polynomials improving previous results by Fox, H. S. Gunaratne, and himself. If \(f_\chi\), the conductor of \(\chi\), is not a \(p\)-power, the results also yield a generalization of formulas for the derivative \(L_p'(0,\chi)\). Finally, for \(f_\chi\) unequal to 1 or \(qp^e\) (\(e\geq 1\)), the author constructs two different extensions of \(L_2(s,t,\chi)\) to values of \(t\) in \(2^{-1}{\mathbb Z}_2\).
scientific article; zbMATH DE number 1902868
scientific article