An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths

Understand

We discover an explicit expansion formula for the powers $s$ of the Euler Product (or Dedekind $\eta$-function) in terms of hook lengths of partitions, where the exponent $s$ is any complex number.

• Several classical formulas have been derived for certain integers $s$ by Euler, Jacobi, Klein, Fricke, Atkin, Winquist, Dyson and Macdonald.
• In particular, Macdonald obtained expansion formulas for the integer exponents $s$ for which there exists a semi-simple Lie algebra of dimension $s$.
• For the type $A_l^{(a)}$ he has expressed the $(t^2-1)$-st power of the Euler Product as a sum of weighted integer vectors of length $t$ for any integer $t$.