On Ramanujan's Reciprocity Theorem and Its Generalizations

摘要

In his lost notebook [R], Ramanujan stated a beautiful reciprocity theorem, which is a generalization of Jacobi's triple product identity. In this paper, we give a proof of a generalized form of this reciprocity theorem. The proof uses Rogers's transformation, or the second iterate of Heine's transformation. Several new reciprocity theorems and their applications are also presented in this paper.