超几何混合型多重正交多项式

中文摘要 AI 辅助

在正实轴上，针对秩为一的 $q\ imes p$ 权重矩阵（其中 $p$ 和 $q$ 为任意正整数），构造了两类超几何混合型多重正交形式：类雅可比（Jacobi-like）系统和类拉盖尔（Laguerre-like）系统。在允许的近对角范围内，两种混合形式均被显式给出。其幂向量分量是单个终止的广义超几何多项式，而其超几何向量分量则是此类多项式的有限和。在类雅可比情形中，这些和化简为在 $(x,1)$ 处取值的终止的 Kampé de Fériet 多项式的有限组合；在混合 beta--Euler 类拉盖尔情形中，它们由一个有限的三角连接系统所支配。对于完整形式，获得了 Gamma 商 Mellin 公式、Meijer $G$ 表示以及 Rodrigues 型公式。在阶梯线上，相关的双正交序列满足具有 $p$ 条次对角线和 $q$ 条超对角线的对偶递推关系，且每个带状系数均由有限的 Gamma--Pochhammer 表达式显式给出。Christoffel 变换和 Gauss--Borel 分解产生递推矩阵的双对角分解。下因子具有封闭的 Pochhammer 公式，而上因子则通过 Christoffel tau 行列式或平移矩子式的交叉比来表示。在混合 Piñeiro 特化中，两条链均显式闭合。合流极限将类雅可比系统与类拉盖尔和类埃尔米特（Hermite-like）系统联系起来。

英文摘要

Two hypergeometric families of mixed-type multiple orthogonal forms are constructed for rank-one $q\times p$ matrices of weights, with arbitrary $p$ and $q$: Jacobi and Laguerre I systems. Both normalized mixed forms are obtained explicitly for admissible near-diagonal multi-indices. The power-vector components are terminating generalized hypergeometric polynomials, while the hypergeometric-vector components are finite sums of such polynomials. In the Jacobi case, these sums are expressed as finite combinations of terminating Kampé de Fériet polynomials evaluated at $(x,1)$. For the mixed beta--Euler Laguerre I system, a finite triangular system relates the residues at finite poles to the terms generated by the Euler operator. Gamma-quotient Mellin formulas, Meijer $G$-representations, and Rodrigues formulas are derived for the complete mixed forms. On the step-line, the two biorthogonal systems satisfy dual recurrences governed by matrices with $p$ subdiagonals and $q$ superdiagonals. All recurrence coefficients are given by finite Gamma--Pochhammer expressions. Under normality and nonvanishing-pivot assumptions, Christoffel transformations and Gauss--Borel factorization yield bidiagonal factorizations of the recurrence matrices. The lower factors have closed Pochhammer formulas, while the upper factors are expressed through finite Christoffel tau-determinants or, equivalently, cross-ratios of shifted moment minors. Both Christoffel chains close explicitly in the mixed Piñeiro specialization.

发表机构

• Complutense University of Madrid（马德里康普顿斯大学）

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