Abstract and Applied Analysis

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 ${l}_{p,q}(s,t,\chi )=\vspace{1pt}{([2]}_{q}/{[2]}_{F}){\sum }_{a=1,(p,a)=1}^{F}{(-1)}^{a}(\chi (a){q}^{a}/{\langle a+pt\rangle }^{s}){\sum }_{m=0}^{\infty }(\begin{smallmatrix}-s\\ \vspace{0pt}m\end{smallmatrix}){(F/\langle a+pt\rangle )}^{m}{E}_{m,{q}^{F}}^{\ast}$ which interpolates the values ${l}_{p,q}(-n,t,\chi )={E}_{n,{\chi }_{n},q}^{\ast}(pt)-{p}^{n}{\chi }_{n}(p)({[2]}_{q}/{[2]}_{{q}^{p}}){E}_{n,{\chi }_{n},{q}^{p}}^{\ast}(t)$, 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.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 360517, 10 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969176

Digital Object Identifier
doi:10.1155/2008/360517

Mathematical Reviews number (MathSciNet)
MR2411043

Zentralblatt MATH identifier
1149.11011

Citation

Kim, Min-Soo; Kim, Taekyun; Park, D. K.; Son, Jin-Woo. On a Two-Variable $p$ -Adic ${l}_{q}$ -Function. Abstr. Appl. Anal. 2008 (2008), Article ID 360517, 10 pages. doi:10.1155/2008/360517. https://projecteuclid.org/euclid.aaa/1220969176

References

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