Remarks on some modular identities

Introduction.We shall consider a certain class of functions invariant with respect to the substitutions of the congruence subgroup T0(p) of the modular group T. By specializing these functions, we shall obtain classical identities in the analytical theory of numbers: E.g., the Ramanujan identities for partitions modulo 5, 7 and Mordell's identity for r(ra).We shall also derive some new identities.These functions bear some resemblance to those considered by Rademacher in his paper [l ](l) to prove the Ramanujan identities, certain modular equations, etc.The type of function considered, however, seems first to have been studied by Watson in his paper [2].