Generalized reciprocal identities

Huber, Tim; Schultz, Daniel

doi:10.1090/proc/13113

Generalized reciprocal identities
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by Tim Huber and Daniel Schultz PDF

Proc. Amer. Math. Soc. 144 (2016), 4627-4639 Request permission

Abstract:

Included in Ramanujan’s Notebooks are two reciprocal identities. The first identity connects the Rogers-Ramanujan continued fraction with an eta quotient. The second identity is a level thirteen analogue. These are special cases of a more general class of relations between eta quotients and modular functions defined by product generalizations of the Rogers-Ramanujan continued fraction. Each identity is shown to be a relation between generators for a certain congruence subgroup. The degree, form, and symmetry of the identities is determined from behavior at cusps of the congruence subgroup whose field of functions the parameters generate. The reciprocal identities encode information about fundamental units and class numbers for real quadratic fields.

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