Infinite families of congruences modulo 5 and 9 for overpartitions

• Autorzy: Gu, C. , Hirschhorn, M. D. , Sellers, J. A. , Xia, E. X. W.
• Języki publikacji: EN

Abstrakty

EN

Let p͞(n) denote the number of overpartitions of n. Recently, a number of congruences modulo 5 and powers of 3 for p͞(n) were established by a number of authors. In particular, Treneer proved that the generating function for p͞(5n) modulo 5 is ∑^∞_n=0 p͞(5n)qⁿ ≡ (q;q)⁶_∞/(q²; q²)³_∞ (mod 5). In this paper, employing elementary methods, we establish the generating function of p͞(5n) which yields the congruence due to Treneer. Furthermore, we prove some new congruences modulo 5 and 9 for p͞(n) by utilizing the fact that the generating functions for p͞(5n) modulo 5 and for p͞(3n) modulo 9 are eigenforms for half-integral weight Hecke operators.

Słowa kluczowe

EN

overpartition; congruence; theta functions

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 1
• Strony: 31--44
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Gu, C.

• Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, P.R. China,

autor

Hirschhorn, M. D.

• School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia,

autor

Sellers, J. A.

• Department of Mathematics, Penn State University, University Park, PA 16802, U.S.A.,

autor

Xia, E. X. W.

• Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, P.R. China,

Bibliografia

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• [3] W. Y. C. Chen, Q. H. Hou, L. H. Sun and L. Zhang, Ramanujan-type congruences for overpartitions modulo 16, Ramanujan J. 40 (2016), 311-322.
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• [18] S. Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc. London Math. Soc. 93 (2006), 304-324.
• [19] L. Q. Wang, Another proof of a conjecture by Hirschhorn and Sellers on overpartitions, J. Integer Seq. 17 (2014), art. 14.9.8.
• [20] E. X. W. Xia, Congruences modulo 9 and 27 for overpartitions, Ramanujan J. 42 (2017), 301-323.
• [21] E. X. W. Xia and O. X. M. Yao, New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions, J. Number Theory 133 (2013), 1932-1949.
• [22] X. Yang, S. P. Cui, and B. L. S. Lin, Overpartition function modulo powers of 2, Ramanujan J. 44 (2017), 89-104.
• [23] O. X. M. Yao, Congruences modulo 64 and 1024 for overpartitions, Ramanujan J., to appear.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).

Identyfikatory

DOI

10.4064/ba8129-9-2017