pentagonal number theorem
where the two sides are regarded as formal power series over .
Proof: For , denote by the coefficient of in the product on the left, i.e. write
By this definition, we have for all
where (resp. ) is the number of partitions of as a sum of an even (resp. odd) number of distinct summands. To fix the notation, let be set of pairs where is a natural number and is a decreasing mapping such that . The cardinal of is thus , and is the union of these two disjoint sets:
Now on the right side of (1) we have
Therefore what we want to prove is
For we have
If , define by
If , define by
In both cases, is decreasing and . The mapping maps takes an element having odd to an element having even , and vice versa. Finally, the reader can verify that . Thus we have constructed a bijection , proving (3).
Now suppose that for some (perforce unique) . The above construction still yields a bijection between and excluding (from one set or the other) the single element :
as in (4). Likewise if , only this element is excluded:
The theorem was discovered and proved by Euler around 1750. This was one of the first results about what are now called theta functions, and was also one of the earliest applications of the formalism of generating functions.
The above proof is due to F. Franklin, (Comptes Rendus de l’Acad. des Sciences, 92, 1881, pp. 448-450).
|Title||pentagonal number theorem|
|Date of creation||2013-03-22 13:57:51|
|Last modified on||2013-03-22 13:57:51|
|Last modified by||bbukh (348)|