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Some Observations on
Alternating Power Sums Laala Khaldi
LIMPAF Laboratory
Department of Mathematics
University of Bouira
10000 Bouira
Algeria
l.khaldi@univ-bouira.dz

Rachid Boumahdi
National Higher School of Mathematics
Sidi Abdellah, Algiers
Algeria
r_boumehdi@esi.dz

in

Abstract:

Let $x\geq1$ be a real number and $T_{n}(m)=-1^{m}+2^{m}-\cdots+(-1)^{n}n^{m}$, where $n$ and $m$ are nonnegative integers with $n\geq1$. In this note we obtain an explicit formula for $T_{\lfloor x\rfloor}(m)$, where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, and we establish a new expression for alternating power sums $T_{n}(m)$ in terms of Stirling numbers of the second kind. Moreover, we give a congruence involving alternating sums of falling factorial.