Some Notes on Alternating Power Sums of Arithmetic Progressions

Journal of Integer Sequences, Vol. 21 (2018), Article 18.7.8

Some Notes on Alternating Power Sums of Arithmetic Progressions

András Bazsó
Institute of Mathematics
University of Debrecen
and
MTA-DE Research Group "Equations Functions and Curves"
Hungarian Academy of Sciences and University of Debrecen
P. O. Box 400
H-4002 Debrecen
Hungary
István Mező
Department of Mathematics
Nanjing University of Information Science and Technology
No. 219 Ningliu Rd.
Pukou, Nanjing, Jiangsu
PR China

Abstract:

We show that the alternating power sum

$\begin{displaymath}r^n - \left(m+r\right)^n + \left(2m+r\right)^n - \cdots + (-1)^{\ell-1} \left(\left(\ell-1\right)m + r\right)^n \end{displaymath}$

can be expressed in terms of Stirling numbers of the first kind and r-Whitney numbers of the second kind. We also prove a necessary and sufficient condition for the integrality of the coefficients of the polynomial extensions of the above alternating power sum.

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(Concerned with sequence A144845.)

Received April 25 2017; revised versions received May 1 2017; September 3 2018; September 4 2018. Published in Journal of Integer Sequences, September 9 2018.

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