© T. Kim et al. 2010
Received: 23 August 2010
Accepted: 30 September 2010
Published: 11 October 2010
Let be a fixed prime number. Throughout this paper, , , , and will, respectively, denote the ring of -adic rational integer, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of natural numbers and .
(see ). Recently, many authors have studied in the different several areas related to -theory (see [1–13]). In this paper, we present a systemic study of some families of multiple Carlitz's type -Bernoulli numbers and polynomials by using the integral equations of -adic -integrals on . First, we derive some interesting equations of -adic -integrals on . From these equations, we give some interesting formulae for the higher-order Carlitz's type -Bernoulli numbers and polynomials in the -adic number field.
By (2.2), we have the following lemma.
By (2.5) and (2.7), one obtains the following theorem.
By (2.13) and (2.14), one obtains the following theorem.
By (2.17), one obtains the following theorem.
By (2.23), (2.25), and (2.26), one obtains the following theorem.
By (2.29), one obtains the following theorem.
Therefore, one obtains the following corollary.
By (2.35) and (2.36), one obtains the following corollary.
Thus, one obtains the following theorem.
By (2.46), (2.48), and (2.50), one obtains the following theorem.
Therefore, one obtains the following theorem.
The authors express their gratitude to The referees for their valuable suggestions and comments. This paper was supported by the research grant of Kwangwoon University in 2010.
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