Mathematical Content!

While looking into symmetric functions, I came across the Newton-Girard Formulas. That page references Raymond Séroul's "Programming for Mathematicians," which seems to be an incredible resource I will surely borrow from the library more often. However, the proof of these formulas (presented on pages 278-279) is incredibly beautiful.

First, start with a function f(T) = (1-X_1*T)(1-X_2*T)...(1-X_n^T). Now, differentiate in two ways: first by expanding out the polynomial and then differentiating, or by using the product rule directly. The second method results in a summation of functions that can be represented by formal power series. By using the fact that these formulas are equal, we subtract the functions and get that the coefficients of the resulting series are zero. Thus, the formulas are simultaneously proven without induction or any case garbage.

This will be a proof for me to write formally some time and memorize for later use. The technique is so wonderful, it makes me smile.