On a Two-Variable p -Adic l q -Function

Abstract

We prove that a two-variable $p$-adic $l_q$-function has the series expansion $l_{p,q}(s,t,\chi )={\left(\right[2]}_q/{\left[2\right]}_F\left){\sum }_{a=1,(p,a)=1}^F{(-1)}^a\right(\chi \left(a\right)q^a/{\langle a+pt\rangle }^s){\sum }_{m=0}^{\infty }\left(\begin{array}{c}-s\\ m\end{array}\right){(F/\langle a+pt\rangle )}^m{E}_{m,q^F}^{*}$ which interpolates the values $l_{p,q}\left(-n,t,\chi \right)=E_{n,{\chi }_n,q}^{\ast }\left(pt\right)-p^n{\chi }_n\left(p\right)({\left[2\right]}_q/{\left[2\right]}_{q^p})E_{n,{\chi }_n,q^p}^{\ast }\left(t\right)$, whenever $n$ is a nonpositive integer. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given in this note is due to Washington.