Bidiagonal Factorization of Banded Recursion Matrices for Mixed-Type Multiple Orthogonal Polynomials
Amílcar Branquinho, Ana Foulquié-Moreno, Manuel Mañas
Abstract
Given a banded matrix $\mathscr{T}_N$ with $p$ subdiagonals and $q$ superdiagonals arising from the Gauss--Borel factorization $\mathscr{M}_N = \mathscr{L}_N^{-1}\mathscr{U}_N^{-1}$ of a moment matrix, this paper constructs explicitly its bidiagonal factorization \[ \mathscr{T}_N = L_1 \cdots L_p\, U_q \cdots U_1. \] Bidiagonal factorizations of this type are central to the study of oscillatory banded matrices and to the spectral Favard theorem for multiple orthogonal polynomials The factorization is obtained via Christoffel transformations of the moment matrix. Provided that the perturbed moment matrices $\mathscr{M}_{N,(b,0)}$ and $\mathscr{M}_{N,(0,a)}$ admit a Gauss--Borel factorization, each bidiagonal factor is a quotient of the corresponding Gauss--Borel factors: \[ U_b = \mathscr{U}_{N,(b,0)}^{-1}\mathscr{U}_{N,(b-1,0)}, \qquad L_a = \mathscr{L}_{N,(0,a-1)}\mathscr{L}_{N,(0,a)}^{-1}. \] Explicit Christoffel-type formulas for the entries of the bidiagonal factors are then derived in terms of certain tau-determinants evaluated at the origin: \[ U_{b,n} = -\frac{τ^B_{b-1,n}\,τ^B_{b,n+1}} {τ^B_{b-1,n+1}\,τ^B_{b,n}}, \qquad L_{a,n+1} = -\frac{τ^A_{a-1,n+2}\,τ^A_{a,n}} {τ^A_{a-1,n+1}\,τ^A_{a,n+1}}. \] As an illustration, the theory is applied to the recurrence matrices of multiple Hahn orthogonal polynomials. For two weights the tetradiagonal case is handled via contiguous hypergeometric relations; for three weights, i.e. the pentadiagonal case, the direct hypergeometric representations are required. In both cases fully explicit bidiagonal factorizations are obtained.