A connection between the theories of restricted partitions into parts congruent to 0 modulo m and unrestricted partitions is constructed here. According to this connection some congruence properties for restricted partitions are given and famous theorems are reformulated. In the first place, the author proves that the number of partitions of mn into parts congruent to 0 modulo m is equal to the number of partitions of n. In addition, a generalization of Euler’s pentagonal number theorem is easily expounded as a special case of Jacobi’s identity. This result is again proved elementary by developing a combinatorial argument due to Franklin. |
