Partition identities and congruences associated with the Fourier coefficients of the Euler products. (Q1414327)

The authors give two applications of the operator \(U(m)\), \(m> 0\) an integer, defined on a formal power series \(\sum^\infty_{n=0} a_n q^n\) as \[ \sum^\infty_{n=0} a_n q^n|_{U(m)}= \sum^\infty_{n=0} a_{mn} q^n. \] They use the relation \((\sum^\infty_{n=0} b_n q^{mn} \sum^\infty_{n=0} a_n q^n)|_{U(m)}= \sum^\infty_{n=0} b_nq^n \sum^\infty_{n=0} a_{mn} q^n\) to give a new proof of Ramanujan's famous congruences \(p(5n+ 4)\equiv 0\text{\,mod\, 5}\), \(p(7n+ 5)\equiv 0\text{\,mod\,}7\) and \(p(11n+ 6)\equiv 0\text{\,mod\,} 11\) where \(p(n)\) denotes the number of unrestricted partitions of the nonnegative integer \(n\). They also use the operator \(U(m)\) to establish other partition Ramanujan identities.