pentagonal number theorem

Theorem :

k=1(1-xk)=n=-(-1)nxn(3n+1)/2 (1)

where the two sides are regarded as formal power series over .

Proof: For n0, denote by f(n) the coefficient of xn in the productPlanetmathPlanetmath on the left, i.e. write


By this definition, we have for all n


where e(n) (resp. d(n)) is the number of partitionsMathworldPlanetmathPlanetmath of n as a sum of an even (resp. odd) number of distinct summands. To fix the notation, let P(n) be set of pairs (s,g) where s is a natural numberMathworldPlanetmath >0 and g is a decreasing mapping {1,2,,s}+ such that xg(x)=n. The cardinal of P(n) is thus f(n), and P(n) is the union of these two disjoint sets:

E(n)={(s,g)P(n)s is even},
D(n)={(s,g)P(n)s is odd}.

Now on the right side of (1) we have


Therefore what we want to prove is

e(n) = d(n)+(-1)m  if n=m(3m±1)/2 for some m (2)
e(n) = d(n)  otherwise. (3)

For m1 we have

m(3m+1)/2 = 2m+(2m-1)++(m+1) (4)
m(3m-1)/2 = (2m-1)+(2m-2)++m (5)

Take some (s,g)P(n), and suppose first that n is not of the form (4) nor (5). Since g is decreasing, there is a unique integer k[1,s] such that

g(j) =g(1)-j+1  for j[1,k],g(j) <g(1)-j+1  for j[k+1,s].

If g(s)k, define g¯:[1,s-1]+ by

g¯(x)={g(x)+1,if x[1,g(s)],g(x),if x[g(s)+1,s-1].

If g(s)>k, define g¯:[1,s+1]+ by

g¯(x)={g(x)-1,if x[1,k],g(x),if x[k+1,s],k, if x=s+1.

In both cases, g¯ is decreasing and xg¯(x)=n. The mapping gg¯ maps takes an element having odd s to an element having even s, and vice versa. Finally, the reader can verify that g¯¯=g. Thus we have constructed a bijection E(n)D(n), proving (3).

Now suppose that n=m(3m+1)/2 for some (perforce unique) m. The above construction still yields a bijection between E(n) and D(n) excluding (from one set or the other) the single element (m,g0):

g0(x)=2m+1-x  for x[1,m]

as in (4). Likewise if n=m(3m-1)/2, only this element (m,g1) is excluded:

g1(x)=2m-x  for x[1,m]

as in (5). In both cases we deduce (2), completing the proof.

Remarks: The name of the theorem derives from the fact that the exponents n(3n+1)/2 are the generalized pentagonal numbersMathworldPlanetmath.

The theorem was discovered and proved by Euler around 1750. This was one of the first results about what are now called theta functionsDlmfMathworld, 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 theoremMathworldPlanetmath
Canonical name PentagonalNumberTheorem
Date of creation 2013-03-22 13:57:51
Last modified on 2013-03-22 13:57:51
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 7
Author bbukh (348)
Entry type Theorem
Classification msc 11P81
Classification msc 14K25