Arithmetic properties of Ramanujan's general partition function for modulo 11
DOI:
https://doi.org/10.48129/kjs.v48i1.8827Keywords:
General partitions, theta-functions, $q$-identities.Abstract
In the present work, for the general partition function $p_k(n)$, we establish five new infinite families of congruences. Our emphasis throughout this paper is to exhibit the use of $q$-identities to generate congruences for the general partition function.References
bibitem[Andrews, 1976]{ge}
textbf{Andrews, G.E. (1976)} in: G.C. Rota(Ed.). The Theory of Partitions.
Encyclopedia of Mathematics and its Applications, 2,
Addison-Wesley, Reading,
(Reprinted: Cambridge Univ. Press, London and New York, 1998): xii+272.
bibitem[Atkin, 1968]{at}
textbf{Atkin, A.O.L. (1968)} Ramanujan congruences for $p_k(n)$. Canad. J. Math. 20: 67-78.
bibitem[Baruah emph{et al.}, 2011]{bo}
textbf{Baruah, N.D. & Ojah, K.K. (2011)} Some congruences deducible from Ramanujan's cubic continued fraction. Int. J. Number Theory, 7: 1331-134.
bibitem[Baruah and Sarmah, (2013)]{bs}
textbf{Baruah, N.D. & Sarmah, B.K. (2013)} Identities and congruences for the general partition and Ramanujan $tau$ functions. Indian J. of pure and Appl. Math. 44(5): 643-671.
bibitem[Berndt, (1991)]{be1}
textbf{Berndt, B.C. (1991)} Ramanujan's Notebooks, Part III. Springer: viii+510.
bibitem[Boylan, (2004)]{mb}
textbf{Boylan, M. (2004)} Exceptional congruences for powers of the partition functions. Acta Arith. 111: 187-203.
bibitem[Chen emph{et. al.} (2014)]{ch}
textbf{Chen, W.Y.C., Du, D.K., Hou, Q.H. & Sun, L.H. (2014)} Congruences of multi-partition functions modulo powers of primes. Ramanujan J. 35: 1-19.
bibitem [Farkas and Kra, (1999)]{fk}
textbf{Farkas, H.M.& Kra, I. (1999)} Ramanujan Partition identities. Contemporary Math. 240:
-130.
bibitem[Gandhi, (1963)]{ga}
textbf{Gandhi, J.M. (1963)} Congruences for $p_k(n)$ and Ramanujan's $tau$ function.
Amer. Math. Mon. 70: 265-274.
bibitem[Gordon, (1983)]{go}
textbf{Gordon, B. (1983)} Ramanujan congruences for $p_kpmod{11^r}$. Glasgow Math. J. 24: 107-123.
bibitem[Hammond and Lewis, (2004)]{ha}
textbf{Hammond, P. & Lewis, R. (2004)}
Congruences in ordered pairs of partitions. Int. J. Math. Math. Sci. 45-48: 2509-2512.
bibitem[Kiming and Olsson, (1992)]{ko}
textbf{Kiming, I. & Olsson, J.B. (1992)} Congruences like Ramanujan's for powers of the partition function. Arch. Math. (Basel) 59(4): 348-360.
bibitem[Newmann, (1995)]{ne}
textbf{Newmann, M. (1955)} An identity for the coefficients of certain modular forms.
J. Lond. Math. Soc. 30: 488-493.
bibitem[Newmann, (1957)]{ne1}
textbf{Newmann, M. (1957)} Congruence for the coefficients of modular forms and some new congruences for the partition function. Canad. J. Math. 9: 549-552.
bibitem[Newmann, (1957)]{ne2}
textbf{Newmann, M. (1957)} Some theorems about $p_k(n)$. Canad. J. Math. 9: 68-70.
bibitem[Ramanathan, (1950)]{rm}
textbf{Ramanathan, K.G. (1950)} Identities and congruences of the Ramanujan type. Canad. J. Math. 2: 168-178.
bibitem[Ramanujan, (1919)]{ra}
textbf{Ramanujan, S. (1919)} Some properties of $p(n),$ the number of partitions of $n$. Proc. Cambridge Phillos. Soc. 19: 207--210.
bibitem[Ramanujan, (1920)]{ra1}
textbf{Ramanujan, S. (1920)} Congruence properties of partitions. Proc. Lond. Math. Soc. 18
Records of 13 March 1919.
bibitem[Ramanujan, (1921)]{ra2}
textbf{Ramanujan, S. (1921)} Congruence properties of partitions. Math. Z. IX: 147-153.
bibitem[Saikia and Chetry, (2018)]{sc}
textbf{Saikia, N. & Chetry, J. (2018)} Infinite families of congruences modulo $7$ for Ramanujan's general partition function. Ann. Math. Quebec. 42(1): 127-132.
bibitem[Tang, (2018)]{ta} textbf{Tang, D. (2018)} Congruences modulo powers of $5$ for $k$-colored partitions.
J. Number Theory, http://doi.org/10.1016/j.jnt.2017.10.027.