×

Ramanujan’s cubic continued fraction and Ramanujan type congruences for a certain partition function. (English) Zbl 1205.11111

Define \(a(n)\) by \[ \sum_{n \geq 0} a(n)q^n := \prod_{n \geq 1} \frac{1}{(1-q^n)(1-q^{2n})}. \] The author proves that for \(k \geq 1\) and \(n \geq 0\), \[ a(3^kn+c_k) \equiv 0 \pmod{3^{k+\delta(k)}}, \] where \(c_k\) is the reciprocal of \(8\) modulo \(3^k\) and \(\delta(k) := \frac{1}{2}(1+ (-1)^k)\). The proof follows classical methods used to establish Ramanujan’s partition congruences \[ p(5^kn+r_k) \equiv 0 \pmod{5^k}, \] where \(p(n)\) is the number of partitions of \(n\) and \(r_k\) is the the reciprocal of \(24\) modulo \(5^k\).

MSC:

11P83 Partitions; congruences and congruential restrictions
11F20 Dedekind eta function, Dedekind sums
11F33 Congruences for modular and \(p\)-adic modular forms
PDFBibTeX XMLCite
Full Text: DOI

Online Encyclopedia of Integer Sequences:

Number of 9-regular cubic partitions of n.

References:

[1] Adiga C., Indian J. Pure Appl. Math. 35 pp 1047–
[2] DOI: 10.1007/s002080000142 · Zbl 1007.11061 · doi:10.1007/s002080000142
[3] DOI: 10.1073/pnas.191488598 · Zbl 1114.11310 · doi:10.1073/pnas.191488598
[4] DOI: 10.1007/s00222-003-0295-6 · Zbl 1038.11067 · doi:10.1007/s00222-003-0295-6
[5] Andrews G. E., Encycl. Math. and Its Appl. 2, in: The Theory of Partitions (1998)
[6] DOI: 10.1017/CBO9781107325937 · doi:10.1017/CBO9781107325937
[7] Andrews G. E., Ramanujan’s Lost Notebook (2005) · Zbl 1075.11001
[8] DOI: 10.1017/S0017089500000045 · Zbl 0163.04302 · doi:10.1017/S0017089500000045
[9] Atkin A. O. L., Proc. London Math. Soc. 18 pp 563–
[10] DOI: 10.1006/jmaa.2001.7823 · Zbl 0997.11037 · doi:10.1006/jmaa.2001.7823
[11] DOI: 10.1016/S0377-0427(03)00612-5 · Zbl 1043.11042 · doi:10.1016/S0377-0427(03)00612-5
[12] DOI: 10.1216/rmjm/1199649823 · Zbl 1133.11035 · doi:10.1216/rmjm/1199649823
[13] DOI: 10.1007/978-1-4612-0965-2 · doi:10.1007/978-1-4612-0965-2
[14] Berndt B. C., Number Theory in the Spirit of Ramanujan (2004) · Zbl 1117.11001
[15] DOI: 10.4153/CJM-1995-046-5 · Zbl 0838.33011 · doi:10.4153/CJM-1995-046-5
[16] Berndt B. C., Acta Arith. 73 pp 67–
[17] DOI: 10.1007/978-3-642-56513-7_3 · doi:10.1007/978-3-642-56513-7_3
[18] Bhargava S., Indian J. Pure Appl. Math. 35 pp 1003–
[19] DOI: 10.1142/S1793042110003150 · Zbl 1203.05008 · doi:10.1142/S1793042110003150
[20] Chan H. H., Acta Arith. 73 pp 343–
[21] DOI: 10.4064/aa126-4-2 · Zbl 1123.11042 · doi:10.4064/aa126-4-2
[22] DOI: 10.1090/conm/254/03956 · doi:10.1090/conm/254/03956
[23] Hirschhorn M. D., Austral. Math. Soc. Gaz. 31 pp 259–
[24] Hirschhorn M. D., J. Reine Angew. Math. 326 pp 1–
[25] Klove T., Acta Arith. 36 pp 219–
[26] Knopp M. I., Modular Functions in Analytic Number Theory (1970) · Zbl 0259.10001
[27] Lovejoy J., J. Reine Angew. Math. 542 pp 123–
[28] DOI: 10.2307/121118 · Zbl 0984.11050 · doi:10.2307/121118
[29] Ono K., The Web of Modularity (2003)
[30] DOI: 10.1142/S1793042110003253 · Zbl 1205.11112 · doi:10.1142/S1793042110003253
[31] Watson G. N., J. Reine Angew. Math. 179 pp 97–
[32] DOI: 10.1016/j.jmaa.2005.07.062 · Zbl 1095.11027 · doi:10.1016/j.jmaa.2005.07.062
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.