Czechoslovak Mathematical Journal, first online, pp. 1-14
5-dissections and sign patterns of Ramanujan's parameter and its companion
Shane Chern, Dazhao Tang
Received May 29, 2020. Published online February 8, 2021.
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 $R(q)R(q^2)^2$ and $R(q)^2/R(q^2)$, 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.
Affiliations: Shane Chern, Department of Mathematics, Penn State University, University Park, McAllister Building, Pollock Rd, State College, PA 16802, USA, e-mail: email@example.com; Dazhao Tang (corresponding author), Center for Applied Mathematics, Tianjin University, No. 92 Weijin Road, Tianjin 300072, P. R. China, e-mail: firstname.lastname@example.org