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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The authors would like to thank Mike Hirschhorn for some helpful suggestions.
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
- sign pattern
- Ramanujan’s parameter