An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality

Abstract

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 $1\le |\mathbf{m}|\le N$, 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.