"COMBINATORIAL IDENTITIES: EULER, GAUSS, JACOBI, MCDONALD, LEIBENZON AND OTHERS.".

Abstract:

In 1760, Euler noticed that if you multiply the
polynomials 1-x, 1-x^2, 1-x^3, 1-x^4, etc., then the
result converges to 1-x-x^2+x^5+x^7-x^12-x^15+x^22+x^26-...
(can you guess the formula? it involves the
"pentagonal numbers" (3n^2+n)/2). This formula has
some unexpected applications, for example,
it provides an easy way to calculate
the number of partitions of a given positive
integer into (unordered) sums of positive integers.
(For example, 5 has 7 such partitions: 5, 4+1, 3+2,
3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; 10 has 42 such
partitions; but how many partitions are there for 20?
for 50? for 100? I will show how to solve these problems
by a two-minute computation; well, you may need three
minutes for 100.)

Some fifty years after Euler's discovery Gauss and Jacobi notices that if you multiply the cubes of the same polynomials (1-x, 1-x^2, etc.), you will get something still more remarkable that Euler's identity. (Take a pencil and a piece of paper and try to calculate and to guess the answer.) This looks especially unexpected, if you notice that the product of squares of our factors does not seem to provide anything interesting. (In 1972, a well known physicist Dyson compiled a list of exponents of the Euler product for which there was a known (to him) formula: 1, 3, 6, 8, 10, 14, 15, 21, 24, 26; we will discuss this briefly).

I spoke about this a couple of years ago, but now I want to show to you a very simple and very elegant proof of the Gauss-Jacobi identity found in 1984 by Zinovi Leibenzon. You will like it.

This material is elementary, no special knowledge is required. Well, if you know how to multiply polynomials, it may help.