Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
  • [2] Bruce C. Berndt, Ramanujan's notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903
  • [3] Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM: A study in analytic number theory and computational complexity; A Wiley-Interscience Publication, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. MR 877728
  • [4] Heng Huat Chan and Sen-Shan Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1 (1997), no. 1, 75-90. MR 1607529,
  • [5] H. Cohen, Number theory. Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, Springer, New York, 2007. MR 2312337 (2008e:11001)
  • [6] J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308-339. MR 554399,
  • [7] Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
  • [8] Graham Everest and Thomas Ward, An introduction to number theory, Graduate Texts in Mathematics, vol. 232, Springer-Verlag London, Ltd., London, 2005. MR 2135478
  • [9] T. Horie and N. Kanou, Certain modular functions similar to the Dedekind eta function, Abh. Math. Sem. Univ. Hamburg 72 (2002), 89-117. MR 1941549,
  • [10] Nobuhiko Ishida, Generators and equations for modular function fields of principal congruence subgroups, Acta Arith. 85 (1998), no. 3, 197-207. MR 1627819
  • [11] Winfried Kohnen and Geoffrey Mason, On generalized modular forms and their applications, Nagoya Math. J. 192 (2008), 119-136. MR 2477614
  • [12] Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603
  • [13] Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
  • [14] Morris Newman, Construction and application of a class of modular functions. II, Proc. London Math. Soc. (3) 9 (1959), 373-387. MR 0107629
  • [15] Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $ q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2004. MR 2020489
  • [16] Srinivasa Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904
  • [17] L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. S1-25, no. 1, 318. MR 1576348
  • [18] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original, Kanô Memorial Lectures, 1. MR 1291394

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F11, 11F20, 11R29

Retrieve articles in all journals with MSC (2010): 11F11, 11F20, 11R29

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

American Mathematical Society