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Sums of powers of consecutive \(q\)-integers. (English) Zbl 1069.11009

For any \(k\in\mathbb Z\), define the \(q\)-integer \([k]_q\) as \([k]_q=(q^k-1)/(q-1)\), where \(q\) is an indeterminate. The author computes a formula for the sum \(\sum_{j=0}^{k-1}q^{hj}[j]_q^n\), where \(k,h,n\in \mathbb Z\) and \(k,n>0\). The result is analogous to the classical formula for \(\sum_{j=0}^{k-1}j^n.\)

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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