E709 -- De evolutione potestatis polynomialis cuiuscunque (1 + x + x2 + x3 + x4 + etc.)n
(On the expansion of the power of any polynomial 1 + x + x2 + x3 + x4 + etc.)
Summary:
(based on Jordan Bell's abstract)
In this paper, Euler gives
- the general trinomial coefficient as a sum of binomial coefficients,
- the general quadrinomial coefficient as a sum of binomial and trinomial coefficients,
- the general quintonomial coefficient as a sum of binomial and quadrinomial coefficients, and
- a way to determine the coefficients in any polynomial (1 + x + x2 + ... + xm)n
as a sum of the coefficients of lower-degree polynomials.
Publication:
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Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 12, 1801, pp. 47-57
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Opera Omnia: Series 1, Volume 16, pp. 28 - 40
Documents Available:
- Original Publication: E709
- English translation (Jordan Bell): E709
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