Constructing orthogonal matrix polynomials satisfying differential equations from two different Laguerre weights

Abstract

We produce an (N+2)-parametric family of matrix polynomials (P _n ) _n of size N×N, which are orthogonal with respect to a weight matrix, involving classical Laguerre scalar weights associated with different exponential functions. This family of orthogonal matrix polynomials satisfies a second-order differential equation with differential coefficients (independent of n) that are matrix polynomials F ₂, F ₁, and F ₀ of degree not larger than 2, 1 and 0, respectively. To proceed in depth, we deal with the particular size 2×2. The Rodrigues formula is obtained to provide an explicit expression as well as the three-term recurrence relation for the referred family of polynomials. As a consequence of the weight's structure, these recurrence coefficients do not behave asymptotically as multiples of the identity.

Keywords:

• orthogonal polynomials
• matrix orthogonality
• differential equations

Acknowledgements

The work of the first author is partially supported by MCI ref. MTM2009-12740-C03-02, FQM-262, FQM-4643 (Junta de Andalucía). This work was prepared during the visit of the second author to the University of Seville.