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The most concise recurrence formula for the sums of integer powers

Published online by Cambridge University Press:  20 June 2025

José Luis Cereceda
José Luis Cereceda*
Affiliation:
C/La Fragua 11, Bajo A, Collado Villalba, 28400 Madrid, Spain e-mail: jl.cereceda@movistar.es
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For integers n, , let Sk (n) denote the power sum 1k + 2k +… + nk, with S0 (n) = n. As is well known, Sk (n) can be represented by a polynomial in n of degree k + 1 with constant term equal to zero, i.e. ${S_k}\,(n)\, = \,\mathop \sum \nolimits_{j\, = \,1}^{k\, + \,1} \,{a_{k,j}}{n^j}$, for certain rational coefficients ak,j. In 1973, in the popular science magazine Kvant*, appeared an article by Vladimir Abramovich [1] in which, among other things, the author derived the following minimal recurrence relation which expresses in a very compact way the connection between the power sums Sk (n) and Sk-1 (n):

(1)

where the polynomial $S_{k\, - \,1}^*\,(n)$ is constructed by replacing in Sk-1 (n) the term nj with the expression ${{{n^{j + 1}} - n} \over {j + 1}}$ for each j = 1, 2, … , k. Note that $S_{k\, - \,1}^*\,(n)$ has degree k + 1, as it should be.

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Information
The Mathematical Gazette , Volume 109 , Issue 575 , July 2025 , pp. 253 - 258
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© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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