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Generalized reciprocal identities

Authors: Tim Huber and Daniel Schultz
Journal: Proc. Amer. Math. Soc. 144 (2016), 4627-4639
MSC (2010): Primary 11F11; Secondary 11F20, 11R29
Published electronically: May 3, 2016
MathSciNet review: 3544515
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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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Additional Information

Tim Huber
Affiliation: School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, Texas 78539

Daniel Schultz
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802

Received by editor(s): January 13, 2016
Published electronically: May 3, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society

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