移位雅可比多项式与Delannoy数
Shifted Jacobi polynomials and Delannoy numbers
摘要：
我们用移位的雅可比多项式表示Delannoy数的加权推广。我们的公式的专业化扩展之间的关系的中心Delannoy数和勒让德多项式，观察到超过50年前，所有的Delannoy数和某些雅可比多项式。另一个特殊化是给出了移位雅可比多项式的加权格路枚举模型，利用这个模型给出了带自然数参数的雅可比多项式正交性的一个新的组合证明.证明涉及正交性的这些多项式的正交性（广义）拉盖尔多项式，因为它们出现在理论的车多项式。本文给出了第一参数为零、第二参数为负整数的雅可比多项式的正交性的组合证明。我们观察到，对于一个奇数第二参数的一个多项式可能会被添加到这个有限的正交多项式序列比什么是由Askey和Romanovski的经典结果预测。序列中剩余的变换后的雅可比多项式或者等于已经列出的Romanovski-雅可比多项式，或者等于具有正的第二参数的类似变换后的雅可比多项式的单项式倍数。我们提供了一个类似的加权广义的薛德数的表达式的雅可比多项式，并使用这个模型，连同曼苏尔和孙的结果，表示Narayana多项式的移位雅可比多项式。
英文摘要：
We express a weighted generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago, to all Delannoy numbers and certain Jacobi polynomials. Another specialization provides a weighted lattice path enumeration model for shifted Jacobi polynomials, we use this to present a new combinatorial proof of the orthogonality of Jacobi polynomials with natural number parameters. The proof relates the orthogonality of these polynomials to the orthogonality of (generalized) Laguerre polynomials, as they arise in the theory of rook polynomials. We provide a combinatorial proof for the orthogonality of certain Romanovski-Jacobi polynomials with zero first parameter and negative integer second parameter, considered as an initial segment of the list of similarly transformed Jacobi polynomials with the same parameters. We observe that for an odd second parameter one more polynomial may be added to this finite orthogonal polynomial sequence than what was predicted by a classical result of Askey and Romanovski. The remaining transformed Jacobi polynomials in the sequence are either equal to the already listed Romanovski-Jacobi polynomials or monomial multiples of similarly transformed Jacobi polynomials with a positive second parameter. We provide expressions for an analogous weighted generalization of the Schröder numbers in terms of the Jacobi polynomials, and use this model, together with a result of Mansour and Sun, to express the Narayana polynomials in terms of shifted Jacobi polynomials.
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