Abstract
In this paper, the author provides an efficient linear recurrence relation for the number of partitions of n into parts not congruent to 0, \(\pm 1\), \(\pm 8\), \(\pm 9\) and \(10 \pmod {20}\). A simple criterion for deciding whether this number is odd or even is given as a corollary of this result. Some results involving overpartitions and partitions into distinct parts have been derived in this context.
Similar content being viewed by others
References
G.E. Andrews, K. Eriksson, Integer Partitions (Cambridge University Press, Cambridge, 2004)
G.E. Andrews, The hard-hexagon model and Rogers-Ramanujan type identities. Proc. Natl. Acad. Sci. USA 78(9), 5290–5292 (1981)
G.E. Andrews, Multiple series Rogers-Ramanujan type identities. Pac. J. Math. 114(2), 267–283 (1984)
G.E. Andrews, Singular overpartitions. Int. J. Number Theory 11(5), 1523–1533 (2015)
L. Carlitz, M.V. Subbarao, A simple proof of the quintuple product identity. Proc. Am. Math. Soc. 32, 42–44 (1972)
S. Corteel, J. Lovejoy, Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)
M.D. Hirschhorn, J.A. Sellers, Arithmetic relations for overpartitions. J. Comb. Math. Comb. Comput. 53, 65–73 (2005)
B. Kim, A short note on the overpartition function. Discrete Math. 309, 2528–2532 (2009)
J. Lovejoy, Gordon’s theorem for overpartitions. J. Comb. Theory Ser. A 103, 393–401 (2003)
J. Lovejoy, Overpartition theorems of the Rogers-Ramaujan type. J. Lond. Math. Soc. 69, 562–574 (2004)
J. Lovejoy, Overpartitions and real quadratic fields. J. Number Theory 106, 178–186 (2004)
K. Mahlburg, The overpartition function modulo small powers of \(2\). Discrete Math. 286, 263–267 (2004)
N.J.A. Sloane, The on-line encyclopedia of integer sequences. Published electronically at http://oeis.org (2016)
M.V. Subbarao, M. Vidyasagar, On Watson’s quintuple product identity. Proc. Am. Math. Soc. 26, 23–27 (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Merca, M. From a Rogers’s identity to overpartitions. Period Math Hung 75, 172–179 (2017). https://doi.org/10.1007/s10998-016-0180-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-016-0180-x