Journal of Integer Sequences, Vol. 25 (2022), Article 22.5.2

Some Observations on Alternating Power Sums


Laala Khaldi
LIMPAF Laboratory
Department of Mathematics
University of Bouira
10000 Bouira
Algeria

Rachid Boumahdi
National Higher School of Mathematics
Sidi Abdellah, Algiers
Algeria

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.

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(Concerned with sequences A008277 A068424 A122045.)

Received March 8 2022; revised versions received June 1 2022; June 7 2022. Published in Journal of Integer Sequences, June 10 2022.

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