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Sums of products of \(q\)-Bernoulli numbers. (English) Zbl 0986.11010

Let \(p\) be a prime number, \(\mathbb Z_p\) the ring of \(p\)-adic integers, \(\mathbb Q_p\) the field of \(p\)-adic numbers, \(\mathbb C_p\) the completion of the algebraic closure of \(\mathbb Q_p\), \(v _p\) the normalized exponential valuation on \(\mathbb C_p\) with \(|p|= 1/p\), and \(q\) an element of \(\mathbb C_p\) satisfying the inequality \(|q - 1|< p ^{-1/(p-1)}\). Also, let \([x] = (1 - q ^x)/(1 - q)\) whenever \(x \in\mathbb C_p\) and \(|x|\leq 1\). The paper under review defines, for each \(m \in\mathbb N\), an \(m\)-th Carlitz \(q\)-Bernoulli number of order \(k \in\mathbb N\) as the value of the \(k\)-multiple \(p\)-adic \(q\)-integral over \(\mathbb Z_p\) of the function \([x _1 +\dots + x _k] ^m\) by the \(q\)-analogue of the \(p\)-adic invariant measure (when \(k = 1\), it coincides with the ordinary \(m\)-th Carlitz \(q\)-Bernoulli number). The notion of an \(m\)-th \(q\)-Bernoulli polynomial of order \(k\) is introduced in a similar manner. The author obtains explicit formulae for these numbers and polynomials, and also, for sums of products of finitely many Carlitz \(q\)-Bernoulli numbers.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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