Euler's pentagonal number theorem

Remarks

The exponents $k(3k - 1)/2$ for $k = 1, -1, 2, -2, 3, -3, \ldots$ are $1, 2, 5, 7, 12, 15, \ldots$, the generalised pentagonal numbers. Franklin's bijective proof reads the identity as a statement on integer partitions of $m$ into distinct parts. The partitions with an even number of parts and those with an odd number of parts are equinumerous, except when $m = k(3k \pm 1)/2$. In that case the difference, even minus odd, is $(-1)^k$. Franklin's involution moves the smallest part onto the largest parts, or back, and it changes the number of parts by one.

Example: for $m = 5$, the partitions into distinct parts are $5$, $4 + 1$ and $3 + 2$. One has an odd number of parts and two have an even number, so the coefficient of $x^5$ is $2 - 1 = 1$, as in the series.

Multiplying by the generating function for partitions $\prod_n (1 - x^n)^{-1}$ gives a recurrence for the partition function: $$p(m) = \sum_{k \ne 0} (-1)^{k+1}\, p\big(m - k(3k - 1)/2\big),$$ with $p(0) = 1$ and $p(j) = 0$ for $j < 0$. For example, $p(5) = p(4) + p(3) - p(0) = 5 + 3 - 1 = 7$. The theorem is a special case of the Jacobi triple product.

Going deeper

For a student

The product $\prod (1 - x^n)$ is the reciprocal of the partition generating function, so the theorem controls the partition numbers $p(m)$. The recurrence above computes $p(m)$ with about $\sqrt{m}$ terms, which was the standard method for tables of $p(m)$. The bijective proof by Franklin is a model of a sign-reversing involution. The identity is the simplest of a family of product-sum identities that includes the Jacobi triple product and the Rogers–Ramanujan identities.