5-Dissections and Sign Patterns of Ramanujan’s Parameter and Its Companion

Abstract

In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction R(q) and its reciprocal. We obtain the 5-dissections for functions R(q)R(q2)2 and R(q)2/R(q2), which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.

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Acknowledgements

The authors would like to thank Mike Hirschhorn for some helpful suggestions.

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Correspondence to Dazhao Tang.

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The second author was supported by the Postdoctoral Science Foundation of China (No. 2019M661005).

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Chern, S., Tang, D. 5-Dissections and Sign Patterns of Ramanujan’s Parameter and Its Companion. Czech Math J (2021). https://doi.org/10.21136/CMJ.2021.0218-20

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Keywords

  • 5-dissection
  • sign pattern
  • Ramanujan’s parameter

MSC 2020

  • 11F27
  • 30B10