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Xinhua Xiong

  1. Cubic partition modulo powers of 5.

    Ср, 04/28/2010 - 15:01 — SomNambul
    Authors: Xinhua Xiong
    Subjects: Number Theory
    Abstract

    We study the congruences of cubic partition function modulo powers of 5. The
    notion of cubic partitions is introduced by Chan and named by Kim in connection
    with Ramanujan's cubic continued fractions.

    Chan has shown that cubic partition function has several analogous properties
    to the number $p(n)$ of partitions, including the generating function, and
    congruence relations. We generalize the results of Chen-Lin and Xiong on the
    congruences of cubic partition function modulo 5 and 5 to the all powers of 5.

  2. Congruences for an arithmetic function from 3-colored Frobenius partitions.

    Ср, 03/03/2010 - 16:01 — SomNambul
    Authors: Xinhua Xiong
    Subjects: Number Theory
    Abstract

    Let $a(n)$ defined by $\sum_{n=1}^{\infty}a(n)q^n :=
    \prod_{n=1}^{\infty}\frac{1}{(1-q^{3n})(1-q^n)^3}.$ In this note, we prove that
    for every nonnegative integer $n$, a(15n+6) \equiv 0\pmod{5}, a(15n+12) \equiv
    0\pmod{5}. As a corollary, we obtained some results of Ono

  3. Congruences modulo powers of 5 for three-colored Frobenius partitions.

    Втр, 03/02/2010 - 16:01 — SomNambul
    Authors: Xinhua Xiong
    Subjects: Number Theory
    Abstract

    Motivated by a question of Lovejoy \cite{lovejoy}, we show that three-colored
    Frobenius partition functions $\c3$ and related arithmetic fuction $\cc3$
    vanishes modulo some powers of 5 in certain arithmetic progressions.

  4. Ramanujan-Type congruences for cubic partition functions.

    Втр, 03/02/2010 - 16:01 — SomNambul
    Authors: Xinhua Xiong
    Subjects: Number Theory
    Abstract

    The cubic partitions, introduced by Chan and Kim, have generating function
    $\sum_{n=0}^{\infty}a(n)= \frac{1}{(q; q)_{\infty}(q^2; q^2)}.$ In this paper,
    we generalize some results of Chen-Lin, which suggest that $a(n)$ should have
    analogous properties of the ordinary partition function. Specifically, we show
    that for every nonnegative integer $n$, $a(5^4n+547)\equiv 0\pmod{5^2},
    a(7^3n+190)\equiv 0\pmod{7^2}, a(7^3n+288 \equiv 0\pmod{7^2} and
    a(7^3n+337)\equiv 0\pmod{7^2}.$

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