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

Abstract:

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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).