Matrix-Valued Gegenbauer-Type Polynomials

• Published in: Constructive approximation Vol. 46; no. 3; pp. 459 - 487
• Main Authors: Koelink, Erik, de los Ríos, Ana M., Román, Pablo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2017; Springer Nature B.V
• Subjects: Analysis; Chebyshev approximation; Decomposition; Differential equations; Eigenvectors; Functions (mathematics); Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Numerical Analysis; Operators (mathematics); Polynomials; Weighting functions; 33C47; Gegenbauer polynomials; 33C45; Matrix-valued differential operators; Darboux factorization; 33E30; Matrix-valued orthogonal polynomials; Shift operator
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-017-9384-4

We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν > 0 . The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν + 1 . The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν = 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.