Abstract
In his 1984 AMS Memoir, Andrews introduced the family of functions \(c\phi _k(n),\) which denotes the number of generalized Frobenius partitions of \(n\) into \(k\) colors. Recently, Baruah and Sarmah, Lin, Sellers, and Xia established several Ramanujan-like congruences for \(c\phi _4(n)\) relative to different moduli. In this paper, employing classical results in \(q\)-series, the well-known theta functions of Ramanujan, and elementary generating function manipulations, we prove a characterization of \(c\phi _4(10n+1)\) modulo 5 which leads to an infinite set of Ramanujan-like congruences modulo 5 satisfied by \(c\phi _4.\) This work greatly extends the recent work of Xia on \(c\phi _4\) modulo 5.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E.: Generalized Frobenius Partitions, vol. 301. Memoirs of the American Mathematical Society, Providence (1984)
Baruah, N.D., Sarmah, B.K.: Congruences for generalized Frobenius partitions with 4 colors. Discret. Math. 311, 1892–1902 (2011)
Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence (2006)
Eichhorn, D., Sellers, J.A.: Computational proofs of congruences for 2-colored Frobenius partitions. Int. J. Math. Math. Sci. 29, 333–340 (2002)
Garvan, F.G.: Generalizations of Dyson’s Rank. PhD Thesis, Pennsylvania State University (1986)
Garvan, F.G., Sellers, J.A.: Congruences for generalized Frobenius partitions with an arbitrarily large number of colors, INTEGERS 14 (2014), Article A7
Kolitsch, L.W.: An extension of a congruence by Andrews for generalized Frobenius partitions. J. Comb. Theory Ser. A 45, 31–39 (1987)
Kolitsch, L.W.: A congruence for generalized Frobenius partitions with 3 colors modulo powers of 3. In: Analytic Number Theory: Proceedings of a Conference in Honor of Paul T. Bateman, p. 343–348. Birkhauser, Boston, MA, (1990)
Lin, B.L.S.: New Ramanujan type congruence modulo 7 for 4-colored generalized Frobenius partitions. Int. J. Number Theory 10, 637–639 (2014)
Lovejoy, J.: Ramanujan like congruences for three colored Frobenius partitions. J. Number Theory 85, 283–290 (2000)
Ono, K.: Congruences for Frobenius partitions. J. Number Theory 57, 170–180 (1996)
Paule, P., Radu, C.S.: The Andrews–Sellers family of partition congruences. Adv. Math. 230, 819–838 (2012)
Sellers, J.A.: New congruences for generalized Frobenius partitions with 2 or 3 colors. Discret. Math. 131, 367–374 (1994)
Sellers, J.A.: An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions. J. Indian Math. Soc. Special Volume to Commemorate the 125th Birth Anniversary of Srinivasa Ramanujan, pp. 97–103 (2013)
Xia, E.X.W.: A Ramanujan-type congruence modulo 5 for 4-colored generalized Frobenius partitions. Ramanujan J. doi:10.1007/s11139-014-9642-z
Xiong, X.: Congruences modulo powers of 5 for three-colored Frobenius partitions. arXiv:1003.0072 [math NT], 2010
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hirschhorn, M.D., Sellers, J.A. Infinitely many congruences modulo 5 for 4-colored Frobenius partitions. Ramanujan J 40, 193–200 (2016). https://doi.org/10.1007/s11139-014-9652-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-014-9652-x