关于正交多项式的几点注记
A few remarks on orthogonal polynomials
摘要：
已知一个给定的、无限支撑的分布的矩序列，我们很快得到：关于该分布正交的一元多项式$\left\{ p_{n}\right\} _{n\geq 0}$的幂级数展开的系数，$\left\{ p_{n}\right\} _{n\geq 0}$在$p_{j},$ $j\leq n$级数中的展开系数，$\left\{ p_{n}\right\} _{n\geq 0}$族的3项递归的系数的两个序列，- 所谓的“线性化系数”，即$p_{j},$ $j\leq m+n.$ \newline系列中的$% p_{n}p_{m}$的展开系数。相反，假设已知给定正交多项式$\left\{ p_{n}\right\} _{n\geq 0},$族的3项递归的系数的两个序列，我们借助它们表示：$p_{n}$的幂级数展开的系数，$x^{n}$在$p_{j},$的级数中的展开的系数$j\leq n,$矩的分布使得多项式$\left\{ p_{n}\right\} _{n\geq 0}$正交。\newline进一步有两个不同的正交多项式族$\left\{ p_{n}\right\} _{n\geq 0}$和$\left\{ q_{n}\right\} _{n\geq 0}$，并知道它们中的每一个的3项递归序列，我们给出这两个多项式族之间的所谓“连接系数”序列。这是$q_{j},$ $j\leq n.$ \newline系列中$p_{n}$的展开式的系数。我们能够做到这一切是由于特殊的方法，其中我们将正交多项式$\left\{ p_{j}\left( x)\right) \right\} _{j=0}^{n}$的向量处理为向量$\left\{ x^{j}\right\} _{j=0}^{n}$通过一些下三角$(n+1)\times (n+1)$矩阵$\mathbf{Π}_{n}.$的线性变换
英文摘要：
Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n$, two sequences of coefficients of the 3-term recurrence of the family of $\left\{ p_{n}\right\} _{n\geq 0}$, the so called "linearization coefficients" i.e. coefficients of expansion of $% p_{n}p_{m}$ in the series of $p_{j},$ $j\leq m+n.$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0},$ we express with their help: coefficients of the power series expansion of $p_{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n,$ moments of the distribution that makes polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ and $\left\{ q_{n}\right\} _{n\geq 0}$ and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called "connection coefficients" between these two families of polynomials. That is coefficients of the expansions of $p_{n}$ in the series of $q_{j},$ $j\leq n.$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left\{ p_{j}\left( x)\right) \right\} _{j=0}^{n}$ as a linear transformation of the vector $\left\{ x^{j}\right\} _{j=0}^{n}$ by some lower triangular $(n+1)\times (n+1)$ matrix $\mathbf{Π}_{n}.$
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