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\vskip 1cm{\LARGE\bf 
Some Notes on Alternating Power Sums \\
\vskip .1in
of Arithmetic Progressions
}
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\large
Andr\'as Bazs\'o\\
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\\
\href{mailto:bazsoa@science.unideb.hu}{\tt bazsoa@science.unideb.hu}\\
\ \\
Istv\'an Mez\H{o}\\
Department of Mathematics\\ 
Nanjing University of Information Science and Technology\\ 
No. 219 Ningliu Rd. \\
Pukou, Nanjing, Jiangsu\\ PR China\\ 
\href{mailto:istvanmezo81@gmail.com}{\tt istvanmezo81@gmail.com}\\
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\begin{abstract}
We show that the alternating power sum
$$
r^n - \left(m+r\right)^n + \left(2m+r\right)^n - \cdots + (-1)^{\ell-1} \left(\left(\ell-1\right)m + r\right)^n.
$$
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. 
\end{abstract}


\section{Introduction}


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