## 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

### Citations:

Zbl 0364.12015; Zbl 0032.00304
Full Text:

### 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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