IJPAM: Volume 96, No. 3 (2014)

HYPERSUMS OF POWERS OF INTEGERS VIA
THE STOLZ-CESÀRO LEMMA

José Luis Cereceda
Distrito Telefónica
Edificio Este 1
28050 Madrid, SPAIN


Abstract. In this paper, we consider the hypersum polynomial $P_k^{(m)}(n) = \sum_{r=1}^{k+m+1} c_{k,m} ^{r} n^r$, which is a generalization of the polynomial associated with the sums of powers of integers $P_k^{(0)}(n) = 1^k + 2^k + \cdots + n^k$. Using the Stolz-Cesàro lemma, we derive an explicit formula for the set of coefficients $\{c_{k,m}^{r}\}_{r=1}^{k+m+1}$ in terms of $\{c_{k,m-1}^{r}\}_ {r=1}^{k+m}$ and the Bernoulli numbers. This allows us to obtain the $m$-th order hypersum $P_k^{(m)} (n)$ once the $(m-1)$-th order hypersum $P_k^{(m-1)} (n)$ is known. Moreover, we show how to obtain $P_k^{(m)} (n)$ from $P_{k-1}^{(m)}(n)$.

Received: June 24, 2014

AMS Subject Classification: 11B57, 11C08

Key Words and Phrases: sums of powers of integers, hypersum polynomial, Stolz-Cesàro lemma

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DOI: 10.12732/ijpam.v96i3.5 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 96
Issue: 3
Pages: 343 - 351


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