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The two essentially different concepts of “form” and “function” are sometimes confused in elementary text-books, and in this note I set out to stress one or two important formal results, in particular the formal interpretation of the binomial theorem for a negative integral exponent. It has been one specific problem, namely the establishing of the Wronski relations
a1-h1=0, a2-a1h1+h2=0, .. ,ak-ak-1h1+...+(-1)khk=0,...
connecting the elementary symmetric functions ar and the homogeneous prodcut sums hr, which has recently focused my attention on this subject and prompted me to write down a few observations.
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- Copyright © Mathematical Association 1957
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page note 91 * The two uses of the sign + though at first sight unfortunate will not cause confusion.
page note 93 * We exclude the case that a′0 + a′1x + … + a′m%m = 0.
page note 94 * See for example Hardy’s “Pure Mathematics” pp. 290, 386. We must assume in addition that every power series which occurs has a positive radius of convergence.
page note 94 † The two polynomial functions x2 -x + 1, 1 with coefficients in the field consisting of the two elements 0, 1 are equal in this field but they are formally different.