Hahn 型多重正交性的 Askey 型合流方案
An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality
- Complutense University of Madrid（马德里康普顿斯大学）
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中文总结 AI 辅助
该研究将 Wolfs 的 Jacobi 型和 Laguerre 型多重正交系统嵌入统一的 Askey 型合流方案，通过 Bernstein 变换构造 Hahn 型祖先，并导出多种正交多项式族及其合流关系。
中文摘要 AI 辅助
我们将 Wolfs 引入的普通 I 型/II 型多重正交性的 Jacobi 型和 Laguerre 型系统嵌入到单一的 Askey 型合流方案中。对 $q$ Jacobi 型权重同时应用 Bernstein 变换，产生一个正有限格 Hahn 型祖先，由多重 beta 积分表示，当 $q=1$ 时退化为经典 Hahn 权重，并收敛回连续系统。参数和尺度极限随后产生 Kravchuk 型、两个 Meixner 型、两个 Charlier 型、Jacobi 型、两个 Laguerre 型和 Hermite 型族。我们通过显式超几何正交性数据实现每个箭头。对于每个近对角多指标 $1\leq |\mathbf m|\leq N$，Hahn 型 II 型多项式是一个终止的 ${}_{q+2}F_{q+1}(1)$ 级数，满足精确的逆 Bernstein 恒等式。我们构造归一化的 I 型形式，并在显式分离和非消失假设下，通过终止超几何函数的有限和恢复其单个多项式分量，并证明唯一性。有限极点项重组成终止的 Kampé de Fériet 块。它们的极限通过在未反射的 Kravchuk 型和 Meixner-II 型分支上在无穷远处进行系数提取和重构获得，在反射分支上通过有限 Lauricella–Horn 扇形和获得。这产生了沿 Kravchuk–Charlier、Meixner–Charlier、Meixner–Laguerre 和 Charlier–Hermite 箭头的扇形分量合流。对于 $q=1$，所有公式都退化为经典族；对于 $q>1$，两个 Meixner 型系统以及两个 Charlier 型系统在归一化行置换下不等价。
英文摘要
We embed the Jacobi-like and Laguerre-like systems for ordinary type-I/type-II multiple orthogonality introduced by Wolfs into a single Askey-type confluence scheme. Applying the Bernstein transform simultaneously to the $q$ Jacobi-like weights produces a positive finite-lattice Hahn-like ancestor, represented by multiple beta integrals, reducing to the classical Hahn weight for $q=1$, and converging back to the continuous system. Parameter and scaling limits then yield Kravchuk-like, two Meixner-like, two Charlier-like, Jacobi-like, two Laguerre-like, and Hermite-like families. We realize every arrow through explicit hypergeometric orthogonality data. For each near-diagonal multi-index $1\leq |\mathbf m|\leq N$, the Hahn-like type-II polynomial is a terminating ${}_{q+2}F_{q+1}(1)$ series satisfying an exact inverse Bernstein identity. We construct the normalized type-I form and, under explicit separation and nonvanishing assumptions, recover its individual polynomial components by finite sums of terminating hypergeometric functions and prove uniqueness. Finite-pole terms regroup into terminating Kampé de Fériet blocks. Their limits are obtained by coefficient extraction and reconstruction at infinity on the unreflected Kravchuk-like and Meixner-II-like branches, and by finite Lauricella--Horn sector sums on the reflected branches. This yields sectorwise component confluence along the Kravchuk--Charlier, Meixner--Charlier, Meixner--Laguerre, and Charlier--Hermite arrows. For $q=1$ all formulas reduce to the classical families; for $q>1$ the two Meixner-like systems, and likewise the two Charlier-like systems, are inequivalent under permutation of normalized rows.
