Pentagonal set of numbers

The pentagonal number theorem by Leonhard Euler is a result from the mathematical sub-area of the function theory and number theory or combinatorics .

The theorem reads: The formal power series in is valid ${\displaystyle q}$

${\displaystyle \prod _{n=1}^{\infty }(1-q^{n})\;=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n(3n-1)/2}=1+\sum _{n=1}^{\infty }(-1)^{n}\left(q^{n(3n+1)/2}+q^{n(3n-1)/2}\right)}$

So the equation is particularly true for complex numbers in the case of absolute convergence, ie . The exponents are for even the pentagonal numbers . The formula is explicit ${\displaystyle q}$ ${\displaystyle |q|<1}$ ${\displaystyle {\tfrac {n(3n-1)}{2}}}$ ${\displaystyle n\geq 1}$

${\displaystyle (1-q)(1-q^{2})(1-q^{3})\cdots \,=1-q-q^{2}+q^{5}+q^{7}-q^{12}\cdots }$

In particular, only the coefficients +1, −1 and 0 appear on the right-hand side (sequence A010815 in OEIS ).

The importance of the pentagonal number theorem for the theory of functions lies in the fact that the left-hand side, up to the factor, is the development of Dedekind's η function . ${\displaystyle q^{1/24}}$ ${\displaystyle q}$

The statement of the Pentagonalzahlensatzes also allows a combinatorial interpretation: It designates the number of number of partitions of in an even number of different summands and the number of paying partitions into an odd number of different summands. Then the -th coefficient of the above series is. ${\displaystyle A_{n}}$ ${\displaystyle n}$ ${\displaystyle B_{n}}$ ${\displaystyle A_{n}-B_{n}}$ ${\displaystyle n}$

The identity of the pentagonal set is a special case of the Jacobi triple product .

Evidence was given by Euler, among others, Carl Gustav Jacobi and combinatorial proof by F. Franklin in 1881 (shown in the number theory textbook by Godfrey Harold Hardy and EM Wright ).

Recursion relations for the partition function

According to Euler, the generating function of the partitions is : ${\displaystyle p(n)}$

${\displaystyle \sum \limits _{n\geq 0}p(n)x^{n}=\prod \limits _{k=1}^{\infty }(1-x^{k})^{-1}}$

or

${\displaystyle \left(\sum _{n=0}^{\infty }p(n)x^{n}\right)\cdot \left(\prod _{n=1}^{\infty }(1-x^{n})\right)=1}$

Development of the infinite product as a power series according to the pentagonal number theorem gives:

${\displaystyle \prod _{n=1}^{\infty }(1-x^{n})=\sum _{n=0}^{\infty }a_{n}x^{n}}$,

where the coefficients follow from the set of pentagonal numbers (they have the values ): ${\displaystyle a_{i}}$ ${\displaystyle 0,1,-1}$

${\displaystyle a_{i}:={\begin{cases}1&{\mbox{ wenn }}i={\frac {1}{2}}(3k^{2}\pm k){\mbox{ und }}k{\mbox{ gerade}}\\-1&{\mbox{ wenn }}i={\frac {1}{2}}(3k^{2}\pm k){\mbox{ und }}k{\mbox{ ungerade}}\\0&{\mbox{ sonst }}\end{cases}}}$

Put in this results

${\displaystyle \left(\sum _{n=0}^{\infty }p(n)x^{n}\right)\cdot \left(\sum _{n=0}^{\infty }a_{n}x^{n}\right)=1}$.

This can also be expressed in such a way that the discrete convolution of the coefficients with the sequence of partition numbers results in one.

With results from comparing the coefficients of the individual powers ${\displaystyle a_{0}\,p(0)=1}$

${\displaystyle \sum _{i=0}^{n}p(n{-}i)a_{i}=0}$

for everyone . From this, the ones from the can be determined recursively. It follows when the term is extracted from the sum and inserted: ${\displaystyle n\geq 1}$ ${\displaystyle p(i)}$ ${\displaystyle a_{j}}$ ${\displaystyle p(n)}$ ${\displaystyle a_{i}}$

${\displaystyle p(n)=\sum _{k\neq 0,g_{k}<n}(-1)^{k-1}p(n-g_{k})}$

with the kth pentagonal number (k can also be negative). The first terms are explicitly: ${\displaystyle g_{k}={\frac {1}{2}}(3k^{2}-k)}$

${\displaystyle p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots }$

These formulas were used by Percy Alexander MacMahon to calculate values ​​of the partition function up to n = 200.

literature

• Hardy, Wright, An introduction to the theory of numbers, Clarendon Press 1975 (Chapter 19: Partitions)

Web links

• English version of Euler's proof in arxiv
• Eric W. Weisstein : Pentagonal Number Theorem . In: MathWorld (English).

Individual evidence

1. ↑ Published in the Abh. The Petersburg Academy for 1780 (published 1783), presented to the Academy by Euler in 1775. Eneström index of Euler's works 541
2. ↑ Hardy, Wright, An introduction to the theory of numbers, Clarendon Press 1975, p. 286