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Victor J. W. Guo

  1. Two truncated identities of Gauss.

    Втр, 05/22/2012 - 15:01 — SomNambul
    Authors: Victor J. W. Guo, Jiang Zeng
    Subjects: Combinatorics
    Abstract

    Two new expansions for partial sums of Gauss' triangular and square numbers
    series are given. As a consequence, we derive a family of inequalities for the
    overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$
    counting the partitions of $n$ with distinct odd parts. Some further
    inequalities for variations of partition function are proposed as conjectures.

  2. Simple proofs of Jensen's, Chu's, Mohanty-Handa's, and Graham-Knuth-Patashnik's identities.

    Втр, 05/18/2010 - 15:01 — SomNambul
    Authors: Victor J. W. Guo
    Subjects: Combinatorics
    Abstract

    Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24],
    we give simple proofs of Jensen's identity $$ \sum_{k=0}^{n}{x+kz\choose
    k}{y-kz\choose n-k} =\sum_{k=0}^{n}{x+y-k\choose n-k}z^k, $$ and Chu's and
    Mohanty-Handa's generalizations of Jensen's identity. We also give a quite
    simple proof of an equivalent form of Graham-Knuth-Patashnik's identity $$
    \sum_{k\geq 0}{m+r\choose m-n-k}{n+k\choose n}x^{m-n-k}y^k =\sum_{k\geq
    0}{-r\choose m-n-k}{n+k\choose n}(-x)^{m-n-k}(x+y)^k, $$ which was
    rediscovered, respectively, by Sun in 2003 and Munarini in 2005.

  3. Factors of binomial sums from the Catalan triangle.

    Чт, 09/03/2009 - 17:38 — SomNambul
    Authors: Victor J. W. Guo, Jiang Zeng
    Subjects: Number Theory
    Abstract

    By using the Newton interpolation formula, we generalize the recent
    identities on the Catalan triangle obtained by Miana and Romero as well as
    those of Chen and Chu. We further study divisibility properties of sums of
    products of binomial coefficients and an odd power of a natural number. For
    example, we prove that for all positive integers $n_1, ..., n_m$,
    $n_{m+1}=n_1$, and any nonnegative integer $r$, the expression
    $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m}
    {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer.

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