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.
Keywords:  5-dissection; sign pattern; Ramanujan's parameter
Classification MSC:  11F27, 30B10
DOI:  10.21136/CMJ.2021.0218-20

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] G. E. Andrews: Ramanujan's "lost" notebook. III: The Rogers-Ramanujan continued fraction. Adv. Math. 41 (1981), 186-208. DOI 10.1016/0001-8708(81)90015-3  | MR 0625893 | Zbl 0477.33009
[2] G. E. Andrews, B. C. Berndt: Ramanujan's Lost Notebook. I. Springer, New York (2005). DOI 10.1007/b13290 | MR 2135178 | Zbl 1075.11001
[3] S. Chern, D. Tang: Vanishing coefficients in quotients of theta functions of modulus five. Bull. Aust. Math. Soc. 102 (2020), 387-398. DOI 10.1017/S0004972720000271 | MR 4176682 | Zbl 07282365
[4] S. Cooper: On Ramanujan's function $k(q)=r(q)r^2(q^2)$. Ramanujan J. 20 (2009), 311-328. DOI 10.1007/s11139-009-9198-5 | MR 2574777 | Zbl 1239.11051
[5] S. Cooper: Level 10 analogues of Ramanujan's series for $1/\pi$. J. Ramanujan Math. Soc. 27 (2012), 59-76. MR 2933486 | Zbl 1282.11032
[6] S. Cooper: Ramanujan's Theta Functions. Springer, Cham (2017). DOI 10.1007/978-3-319-56172-1 | MR 3675178 | Zbl 1428.11001
[7] D. Q. J. Dou, J. Xiao: The 5-dissections of two infinite product expansions. To appear in Ramanujan J. DOI 10.1007/s11139-019-00200-w
[8] J. Frye, F. Garvan: Automatic proof of theta-function identities. Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts and Monographs in Symbolic Computation. Springer, Cham (2019), 195-258. DOI 10.1007/978-3-030-04480-0_10 | MR 3889559
[9] F. Garvan: A $q$-product tutorial for a $q$-series MAPLE package. Sémin. Lothar. Comb. 42 (1999), Article ID B42d, 27 pages. MR 1701583 | Zbl 1010.11072
[10] C. Gugg: Two modular equations for squares of the Rogers-Ramanujan functions with applications. Ramanujan J. 18 (2009), 183-207. DOI 10.1007/s11139-008-9121-5 | MR 2475936 | Zbl 1193.33230
[11] M. D. Hirschhorn: On the expansion of Ramanujan's continued fraction. Ramanujan J. 2 (1998), 521-527. DOI 10.1023/A:1009789012006 | MR 1665326 | Zbl 0924.11005
[12] M. D. Hirschhorn: The Power of $q$: A Personal Journey. Developments in Mathematics 49. Springer, Cham (2017). DOI 10.1007/978-3-319-57762-3 | MR 3699428 | Zbl 06722024
[13] S.-Y. Kang: Some theorems on the Rogers-Ramanujan continued fraction and associated theta function identities in Ramanujan's Lost Notebook. Ramanujan J. 3 (1999), 91-111. DOI 10.1023/A:1009869426750 | MR 1687021 | Zbl 0930.11025
[14] S. Raghavan, S. S. Rangachari: On Ramanujan's elliptic integrals and modular identities. Number Theory and Related Topics. Tata Institute of Fundamental Research Studies in Mathematics 12. Oxford University Press, Oxford (1989), 119-149. MR 1441328 | Zbl 0748.33013
[15] S. Ramanujan: Notebooks of Srinivasa Ramanujan. II. Tata Institute of Fundamental Research, Bombay (1957). DOI 10.1007/978-3-662-30224-8 | MR 0099904 | Zbl 0138.24201
[16] S. Ramanujan: The Lost Notebook and Other Unpublished Papers. Springer, Berlin; Narosa Publishing House, New Delhi (1988). MR 0947735 | Zbl 0639.01023
[17] B. Richmond, G. Szekeres: The Taylor coefficients of certain infinite products. Acta Sci. Math. 40 (1978), 347-369. MR 0515217 | Zbl 0397.10046
[18] L. J. Rogers: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25 (1894), 318-343. DOI 10.1112/plms/s1-25.1.318 | MR 1576348
[19] D. Tang: On 5- and 10-dissections for some infinite products. To appear in Ramanujan J. DOI 10.1007/s11139-020-00340-4
[20] D. Tang, E. X. W. Xia: Several $q$-series related to Ramanujan's theta functions. Ramanujan J. 53 (2020), 705-724. DOI 10.1007/s11139-019-00187-4 | MR 4173465
[21] E. X. W. Xia, A. X. H. Zhao: Generalizations of Hirschhorn's results on two remarkable $q$-series expansions. To appear in Exp. Math. DOI 10.1080/10586458.2020.1712565

Affiliations:   Shane Chern, Department of Mathematics, Penn State University, University Park, McAllister Building, Pollock Rd, State College, PA 16802, USA, e-mail: shanechern@psu.edu; Dazhao Tang (corresponding author), Center for Applied Mathematics, Tianjin University, No. 92 Weijin Road, Tianjin 300072, P. R. China, e-mail: dazhaotang@sina.com


 
PDF available at: