This paper concerns a set of identities of which the following is the simplest:
Here is the Dedekind -function. The paper offers about a dozen identities of this shape, together with a unified method of proof, given in §4: “Let be a modular function with invariance group such that the genus . If there exists a group such that , , and , then we always have an identity of the form
where , and are both polynomials in , and generates the function field of . To determine , we first set . Since by assumption, is a bijection from , we conclude that
Note that is defined to be 1 if . The polynomial can then be determined by comparing the Fourier expansions of and at .”
In practice, is an extension of a level congruence subgroup by an Atkin-Lehner involution , where . This new method of proof yields some modular equations beyond those of Ramanujan.
The final section lists “all genus 0 discrete groups , , where the is generated by together with all its Atkin-Lehner involutions”.