The method of characteristics, and "problem 89" of Graham, Knuth and Patashnik

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

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. Finally we give a mult…

The Graham--Knuth--Patashnik recurrence: Symmetries and continued fractions

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (αn + βk + γ)\, T(n-1,k)+(α' n + β' k + γ') \, T(n-1,k-1)$ with initial condition $T(0,k) = δ_{k0}$ and parameters $\mathbfμ = (α,β,γ, α',β',γ')$. We show that the family of arrays $T(\mathbfμ)$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbfμ \in \mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ is given by a Stieltjes-type continued fraction in $t$ with coef…

对Graham-Knuth-Patashnik递推关系理论中对数凹性和对数凸性的研究

Log-concavity and log-convexity in the theory of the Graham--Knuth--Patashnik recurrences

本文研究了由Graham-Knuth-Patashnik递推关系定义的三角数组T(n,k;μ)，证明了其在变量参数下具有强对数凹性和强对数凸性，并展示了其Hankel-完全正性。

关于 Graham、Knuth、Patashnik 递推关系的极限分布

On limiting distributions of Graham, Knuth, Patashnik recurrences

本文研究 Graham–Knuth–Patashnik 六参数递推关系解所诱导整值随机变量序列的极限分布，并在 α′=0、其余参数非负时完整刻画其极限行为。

Identities in character tables of $S_n$

In the classic "Concrete Math", by Graham, Patashnik and Knuth, it is stated that "The numbers in Pascal's triangle satisfy, practically speaking, infinitely many identities, so it is not too surprising that we can find some surprising relationships by looking closely." The aim of this note is to indicate that a similar statement seems to hold for the character tables of the symmetric groups $S_n$. Just as important, it is a case-study in using a computer algebra system to prove deep identities, way beyond the ability of mere humans. This article is accomanied by a Maple pacgage, Sn, and ample…