On certain identities of Ramanujan
Starting with a Ramanujan identity involving Lambert series related to modular equations of degree 7, the author gives new proofs of various theta function identities previously proved by \textit{B. Berndt} and \textit{L.-C. Zhang} [J. Number Theory 48, 224--242 (1994; Zbl 0812.11028)] using modular forms. The paper also gives a simple proof of one of Ramanujan's identities that implies the congruence \(p(7n+5)\equiv 0\pmod 7\) for the partition function. The author uses theta functions to construct an elliptic function whose poles are known and then notes that the sum of the residues at the poles inside a period parallelogram is zero.
scientific article; zbMATH DE number 1486245
scientific article