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On Carlitz’s \(q\)-Bernoulli numbers. (English) Zbl 0501.12020

Bernoulli numbers and polynomials can be used to define \(p\)-adic analogues of the classical zeta function and \(L\)-functions (as an integral of a simple function with respect to a measure that is a regularization of a Bernoulli distribution, see [the author, \(p\)-adic numbers, \(p\)-adic analysis, and zeta-functions. Graduate Texts in Mathematics. 58. New York etc.: Springer-Verlag (1977; Zbl 0364.12015)]. In this paper the author generalizes the above by starting with \(q\)-extensions of Bernoulli numbers and polynomials defined by L. Carlitz [Duke Math. J. 15, 987–1000 (1948; Zbl 0032.00304)] (here \(q\) is a \(p\)-adic number of the form \(1+t\) where \(| t|_p<p^{1/(1-p)}\)). He obtains \(p\)-adic \(q\)-\(L\)-functions \(L_{p;q}(s,\chi)\) interpolating certain values connected with the \(q\)-Bernoulli numbers, for which \(\lim_{q\to 1} L_{p;q}(s,\chi)\) is the Kubota-Leopoldt \(p\)-adic \(L\)-function. Also generalizations of the well known Kummer congruences are obtained.

MSC:

11S40 Zeta functions and \(L\)-functions
11B68 Bernoulli and Euler numbers and polynomials
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References:

[1] Carlitz, L., q-Bernoulli numbers and polynomials, Duke math. J., 15, 987-1000, (1948) · Zbl 0032.00304
[2] Carlitz, L., q-Bernoulli and Eulerian numbers, Trans. amer. math. soc., 76, 332-350, (1954) · Zbl 0058.01204
[3] Koblitz, N., ()
[4] Koblitz, N., q-extension of the p-adic gamma function, Trans. amer. math. soc., 260, 449-457, (1980) · Zbl 0443.12008
[5] Lang, S., ()
[6] Mazur, B., Analyse p-adique, ()
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