ABSTRACT We investigate the sequence of random polynomials generated by the three-term recurrence relation , , with initial conditions , , assuming that is a sequence of positive i.i.d. random variables. is a sequence of orthogonal polynomials on the real line, and is the characteristic polynomial of a Jacobi matrix . We investigate the relation between the common distribution of the recurrence coefficients and two other distributions obtained as weak limits of the averaged empirical and spectral measures of . Our main result is a description of combinatorial relations between the moments of the aforementioned distributions in terms of certain classes of coloured planar trees. Our approach is combinatorial, and the starting point of the analysis is a formula of P. Flajolet for weight polynomials associated with labelled Dyck paths.

中文翻译：

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关于由对称三项递推关系生成的随机多项式

摘要 我们研究了由三项递推关系 生成的随机多项式序列，具有初始条件 ， ，假设它是一个正 iid 随机变量的序列。是实线上的一系列正交多项式，是雅可比矩阵的特征多项式。我们研究了递归系数的常见分布与作为 的平均经验和谱测量的弱极限获得的其他两个分布之间的关系。我们的主要结果是根据某些类别的彩色平面树描述了上述分布的矩之间的组合关系。我们的方法是组合的，分析的起点是与标记的 Dyck 路径相关的权重多项式的 P. Flajolet 公式。