New proofs of Ramanujan's partition identities for moduli 5 and 7
Let \(|q|< 1\) and \((a; q)_\infty= \prod^\infty_{n=0} (1- aq^n)\). Ramanujan found identities for \(q(q^5; q^5 )^5_\infty/ (q;q)_\infty\) and \(q(q; q)^3_\infty (q^7; q^7)^3_\infty+ 8q^2 (q^7; q^7)^7_\infty/ (q; q)_\infty\). These identities lead to Ramanujan's celebrated congruences \(p(5n+ 4)\equiv 0\bmod 5\) and \(p(7n+ 5)\equiv 0\bmod 7\) where \(p(n)\) denotes the partition function. As to the proofs of the identities, see \textit{W. N. Bailey} [Q. J. Math., Oxf., II. Ser. 3, 29-31 (1952; Zbl 0046.04203) and 158-160 (1952; Zbl 0046.27202)], \textit{N. J. Fine} [Tôhoku Math. J., II. Ser. 8, 149-164 (1956; Zbl 0073.02904)]\ and \textit{S. Raghavan} [Q. J. Math., Oxf., II. Ser. 37, 221-229 (1986; Zbl 0585.10033)]. In the paper under review, new proofs of the identities are given. The common feature of these two proofs is that they begin with the same trigonometric identity. The proof of the first identity is elementary. As to the second identity, also a short and elegant proof by F. Garvan is included.
scientific article; zbMATH DE number 792473