Cooper, Shaun; Ye, Dongxi The level 12 analogue of Ramanujan’s function \(k\). (English) Zbl 1404.11040 J. Aust. Math. Soc. 101, No. 1, 29-53 (2016). Summary: We provide a comprehensive study of the function \(h=h(q)\) defined by \[ h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})} \] and show that it has many properties that are analogues of corresponding results for Ramanujan’s function \(k=k(q)\) defined by \[ k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}. \] Cited in 4 Documents MSC: 11F11 Holomorphic modular forms of integral weight 05A10 Factorials, binomial coefficients, combinatorial functions 33C75 Elliptic integrals as hypergeometric functions 33E05 Elliptic functions and integrals Keywords:Dedekind eta function; Eisenstein series; elliptic integral; hypergeometric function; Jacobi triple product identity; modular equation; quintuple product identity; Ramanujan’s cubic continued fraction; Rogers-Ramanujan continued fraction PDFBibTeX XMLCite \textit{S. Cooper} and \textit{D. Ye}, J. Aust. Math. Soc. 101, No. 1, 29--53 (2016; Zbl 1404.11040) Full Text: DOI References: [1] Mahadeva Naika, South East Asian J. Math. Math. Sci. 10 pp 129– (2012) [2] DOI: 10.2478/s11533-008-0031-y · Zbl 1176.33020 · doi:10.2478/s11533-008-0031-y [3] DOI: 10.1080/00018736000101189 · doi:10.1080/00018736000101189 [4] Dharmendra, Pure Math. Sci. 1 pp 197– (2012) [5] DOI: 10.1007/s10114-011-0202-9 · Zbl 1282.11028 · doi:10.1007/s10114-011-0202-9 [6] DOI: 10.1016/j.aim.2003.07.012 · Zbl 1122.11087 · doi:10.1016/j.aim.2003.07.012 [7] Cooper, J. Ramanujan Math. Soc. 27 pp 59– (2012) [8] DOI: 10.4153/CJM-2011-079-2 · Zbl 1296.33011 · doi:10.4153/CJM-2011-079-2 [9] DOI: 10.1007/s11139-009-9198-5 · Zbl 1239.11051 · doi:10.1007/s11139-009-9198-5 [10] DOI: 10.1090/stml/034 · doi:10.1090/stml/034 [11] Cooper, J. Comb. Number Theory 1 pp 53– (2009) [12] DOI: 10.1007/978-1-4612-0965-2 · doi:10.1007/978-1-4612-0965-2 [13] DOI: 10.1112/plms/pdp007 · Zbl 1248.11031 · doi:10.1112/plms/pdp007 [14] DOI: 10.1142/S1793042106000401 · Zbl 1159.33300 · doi:10.1142/S1793042106000401 [15] DOI: 10.1088/1751-8113/41/20/205203 · Zbl 1152.33003 · doi:10.1088/1751-8113/41/20/205203 [16] DOI: 10.1112/S0025579309000436 · Zbl 1275.11035 · doi:10.1112/S0025579309000436 [17] DOI: 10.1017/CBO9781107325937 · doi:10.1017/CBO9781107325937 [18] DOI: 10.1142/S1793042110002879 · Zbl 1303.11009 · doi:10.1142/S1793042110002879 [19] DOI: 10.1017/S0013091509000959 · Zbl 1223.33007 · doi:10.1017/S0013091509000959 [20] DOI: 10.4153/CMB-2009-050-5 · Zbl 1234.11141 · doi:10.4153/CMB-2009-050-5 [21] DOI: 10.4064/aa124-2-4 · Zbl 1127.11035 · doi:10.4064/aa124-2-4 [22] Venkatachaliengar, Development of Elliptic Functions According to Ramanujan (2012) [23] DOI: 10.1007/s11253-011-0476-1 · Zbl 1240.11025 · doi:10.1007/s11253-011-0476-1 [24] DOI: 10.1007/s11139-007-9040-x · Zbl 1226.11113 · doi:10.1007/s11139-007-9040-x [25] Richmond, Electron. J. Combin. 16 (2009) [26] Ramanujan, Q. J. Math. 45 pp 350– (1914) [27] DOI: 10.1142/S1793042113500723 · Zbl 1335.11008 · doi:10.1142/S1793042113500723 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.