1. Combinatorics

     We introduce the hook length expansion technique and explain how to discover old and new hook length formulas for partitions. In particular, we derive an expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores.

 2. Number Theory

    Our formula unifies the classical Jacobi and Gauss identities when t=2. We proved some arithmetical properties for t=3 and made a general conjecture. In the case t=5, the conjecture implies the long standing Lehmer conjecture which says that the Ramanujan tau-function never takes the zero value.