Generalized Stirling numbers and sums of powers of arithmetic progressions

José Luis Cereceda ^[1]
1. [1] Collado Villalba
Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 51, Nº. 6, 2020, págs. 954-966
Idioma: inglés
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Resumen
- In this paper, we first focus on the sum of powers of the first n positive odd integers, Tk(n) = 1k + 3k + 5k +···+ (2n − 1)k, and derive in an elementary way a polynomial formula for Tk(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.