# Notation for the “binomial form” of a polynomial

In Hardy's A Course of Pure Mathematics (§117 in the 10th edition), in a discussion of differentiation of polynomials, he introduces what he calls the "binomial form" of a polynomial: $$a_0x^n + \pmatrix{n \\ 1}a_1x^{n-1} + \pmatrix{n\\2}a_2x^{n-2} + \ldots + a_n$$ and states that this is often written symbolically as: $$(a_0, a_1, \ldots, a_n)\!\!(x, 1)^n$$ This is a notation that I have not encountered before. I would be very interested in any information about this notation and its history. Also I have one specific question: what does the $$1$$ in $$)\!\!(x, 1)$$ mean?

• This notation was very common in mid to late 1800s math literature (and if it helps, pretty much any paper by Arthur Cayley uses it), almost all of which is freely available at google books. Besides Cayley's papers (e.g. in his Collected Works), try looking through various volumes of The Quarterly Journal of Pure and Applied Mathematics, as I think it was especially used by English mathematicians. I forgot what it means, however. – Dave L Renfro Apr 7 at 18:48
• @DaveLRenfro: many thanks. I think it is clear, e.g., from the first article by Cayley in volume 6 of that journal, that: $(a_0, a_1, \ldots, a_n)\!\!(x, y)^n$ is like $(a_0, a_1, \ldots, a_n)\!\!(x, 1)^n$ but with $x^{n-i}y^i$ in place of $x^{n-i}$ for each $i$. – Rob Arthan Apr 7 at 21:34