关于Boros-Moll多项式的组合数学
On the Combinatorics of the Boros-Moll Polynomials
摘要：
Boros-Moll多项式出现在四次积分的评估中。原来的双重求和公式并不意味着这些多项式的系数是正的。Boros和Moll利用Ramanujan主定理将双和化为单和，证明了正性。基于Mullin和Rota引入的不情愿函数的结构以及Meixner内函数和双色置换之间的Foata双射的扩展，我们找到了正性的组合证明。事实上，从我们的组合论证中，我们可以看到，从本质上讲，正是二项式定理使得将双和简化为单和成为可能。
英文摘要：
The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan's Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata's bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum.
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