  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 $t$. Kostant has considered the general case for any
positive integer $s$ and obtained further properties.
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  The present paper proposes a new approach. We convert the weighted vectors of
length $t$ used by Macdonald in his identity for type $A_l^{(a)}$ to weighted
partitions with free parameter $t$, so that a new identity on the latter
combinatorial structures can be derived without any restrictions on $t$. The
surprise is that the weighted partitions have a very simple form in terms of
hook lengths of partitions. As applications of our formula, we find some new
identities about hook lengths, including the "marked hook formula". We also
improve a result due to Kostant. The proof of the Main Theorem is based on
Macdonald's identity for $A_l^{(a)}$ and on the properties of a bijection
between $t$-cores and integer vectors constructed by Garvan, Kim and Stanton.
