Consider the polynomial sequence p_0 = x^0 p_1 = x^1 p_2 = x^2 + 1x^1 p_3 = x^3 + 3x^2 + 1x^1 p_4 = x^4 + 6x^3 + 7x^2 + 1x^1 p_5 = x^5 + 10x^4 + 25x^3 + 15x^2 + 1x^1 ... where the coefficients here are the Stirling numbers of the second kind. Then {p_n} is of binomial type.
That is, e.g., for n = 4, [p_4](x + y) = [p_4](x) + [p_4](y) + 4[p_3](x)*[p_1](y) + 4[p_1](x)*[p_3](y) + 6[p_2](x)*[p_2](y), and similarly, for all n. That is, [p_n](x + y) is built from [p_i](x), [p_j](y) in the same way that (x + y)^n is from built from x^i, y^j.
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This is by no means the only polynomial sequence of binomial type, apart from x^n itself. On the contrary, the woods are full of them. But this example is, in a sense, primitive; it results from setting equal to 1 all the arbitrary constants that appear in the general case.
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I'm puzzled why this fact isn't quite so famous as I'd say it should be. (At any rate, I've never before seen it mentioned.) It would seem, for example, to provide an alternate definition of the Stirling numbers of the second kind -- which are glamorous combinatorial celebrities.
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