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

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arxiv math/0406620 v2 pith:L3C6VUXW submitted 2004-06-30 math.CO

Herbert S. Wilf This is my paper · ORCID
classification math.CO
keywords characteristicsgrahamknuthmethodpatashnikproblemsequencevanishes
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We apply the method of characteristics for the solution of pde's to two combinatorial problems. The first is finding an explicit form for a distribution that arises in bio-informatics. The second is a question raised by Graham, Knuth and Patashnik abiout a sequence of generalized binomial coefficients. We find an exact formula, which factors in an interesting way, in the case where one of the six parameters of the problem vanishes. We also show that the associated polynomial sequence has real zeros only, provided that one parameter vanishes, and the other five are nonnegative.

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Cited by 4 Pith papers

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

    math.CO 2026-07 accept novelty 6.0 of 10

    GKP arrays T(n,k;µ) are coefficientwise strongly log-concave and their generating polynomials Pn(x;µ) are coefficientwise strongly log-convex (hence Hankel-TP2) when parameters are indeterminates.

  2. Zeros of GKP sequences of polynomials

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    Under |φ_n|+ψ_n<0, GKP polynomials have real simple interlacing zeros between the roots of the driving quadratic, with explicit extreme-zero asymptotics when ψ is constant.

  3. Moments for generalizations of a coin flip game

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    Derives recursive and closed formulas for moments of waiting times for prescribed words in coin flips and die rolls using one-parameter Eulerian number extensions, Goulden-Jackson cluster method, and Faà di Bruno's formula.

  4. Triangular Arrays using context-free grammar

    math.CO 2025-11 unverdicted novelty 5.0 of 10

    Triangular arrays defined by the recurrence T(n,k) = (a2 n + a1 k + a0) T(n-1,k) + (b2 n + b1 k + b0) T(n-1,k-1) are interpreted as increasing trees via the Hao grammar, yielding explicit formulas for r-Whitney-Euleri...

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