AI 中文总结
研究具多项式系数线性齐次递推的多项式序列的Riordan阵列表示，刻画二阶递推情形并推广到任意阶；构建任意正交多项式序列的生成矩阵框架，还将其拓展至$d$-正交多项式序列，揭示对应递推与矩序列等性质。
AI 中文摘要
我们研究满足具多项式系数的线性齐次递推关系的多项式序列$\{p_n(x)\}$何时以普通Riordan阵作为其系数矩阵。针对二阶递推$p_{n+2}(x)=a(x)p_{n+1}(x)+b(x)p_n(x)$，我们给出了完整刻画。我们将这一结论推广到任意阶递推$\ell \geq 3$的情形，给出了一个必要条件以及完整的因子化判据。\n随后，我们为任意正交多项式序列（OPS）构建了生成矩阵框架，该框架并不局限于Riordan阵类型的情形：对于正交多项式序列的任意下三角可逆系数矩阵$A$，其生成矩阵$P_{A^{-1}}=ASA^{-1}$始终是编码三项递推关系的三对角雅可比矩阵，且$A^{-1}$的第一列始终给出矩序列——即便像勒让德多项式这类递推系数并非常数、且系数矩阵不存在Riordan阵表示的情况也依然成立。我们分别以勒让德多项式和切比雪夫多项式作为非Riordan型和Riordan型的实例进行说明，并给出了广义盖根鲍尔-亨伯特正交多项式序列的显式闭式系数矩阵及其逆矩阵。最后，我们将该框架拓展至$d$-正交多项式序列，证明了$d$-正交序列的系数矩阵始终可逆，其关联的生成矩阵为$(d+2)$-带状（下海森伯格）矩阵，且编码了对应的$(d+2)$-项递推关系，逆矩阵的前$d$列可还原定义向量泛函的矩序列——这一结论将$d$-正交性与$(d+1)$-海森伯格广义Riordan阵对应起来，并推广了经典的$d=1$三对角对应关系。
英文摘要
We study when a polynomial sequence $\{p_n(x)\}$ satisfying a linear homogeneous recurrence with polynomial coefficients admits an ordinary Riordan array as its coefficient matrix. For second-order recurrences $p_{n+2}(x)=a(x)p_{n+1}(x)+b(x)p_n(x)$, we give a complete characterization. We extend this to a necessary condition and a full factorization criterion for recurrences of arbitrary order $\ell \geq 3$. We then develop a production-matrix framework for arbitrary orthogonal polynomial sequences (OPS), not restricted to the Riordan-array-type case: for any lower-triangular invertible coefficient matrix $A$ of an OPS, the production matrix $P_{A^{-1}}=ASA^{-1}$ is always the tridiagonal Jacobi matrix encoding the three-term recurrence, and the first column of $A^{-1}$ always gives the moment sequence -- even when, as for the Legendre polynomials, the recurrence coefficients are non-constant and no Riordan array representation of the coefficient matrix exists. We illustrate this with Legendre and Chebyshev polynomials as, respectively, non-Riordan-type and Riordan-type examples, and give an explicit closed-form coefficient matrix and its inverse for the generalized Gegenbauer--Humbert OPS. Finally, we extend the framework to $d$-orthogonal polynomial sequences, showing that the coefficient matrix of a $d$-orthogonal sequence is always invertible, that its associated production matrix is $(d+2)$-banded (lower Hessenberg) and encodes the corresponding $(d+2)$-term recurrence, and that the first $d$ columns of the inverse matrix recover the moment sequences of the defining vector functional -- identifying $d$-orthogonality with $(d+1)$-Hessenberg generalized Riordan arrays and extending the classical $d=1$ tridiagonal correspondence.
