Generalized Stirling numbers and sums of powers of arithmetic progressions. (English) Zbl 1475.97002
Summary: In this paper, we first focus on the sum of powers of the first \(n\) positive odd integers, \(T_k(n) = 1^k + 3^k + 5^k + \cdots +(2 n - 1)^k\), and derive in an elementary way a polynomial formula for \(T_k(n)\) in terms of a specific type of generalized Stirling numbers. Then we consider the sum of powers of an arbitrary arithmetic progression and obtain the corresponding polynomial formula in terms of the so-called \(r\)-Whitney numbers of the second kind. This latter formula produces, in particular, the well-known formula for the sum of powers of the first \(n\) natural numbers in terms of the usual Stirling numbers of the second kind. Furthermore, we provide several other alternative formulas for evaluating the sums of powers of arithmetic progressions.
MSC:
| 97F60 | Number theory (educational aspects) |
| 11B73 | Bell and Stirling numbers |
| 11B68 | Bernoulli and Euler numbers and polynomials |
Keywords:
sums of powers of integers; polynomial basis; arithmetic progressions; generalized Stirling numbers; Whitney numbers; general binomial coefficientsSoftware:
OEISReferences:
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