Abstract
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 nu >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 nu and nu +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 nu =1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.
Highlights
Matrix-valued orthogonal polynomials have been introduced and studied by Krein in connection with spectral analysis and moment problems, see [8,33,34]
The first case was studied by Grünbaum et al [23] for the case of the symmetric pair (G, K ) = (SU(3), U(2)) relying heavily on invariant differential operators
[29,30] give a detailed study of the matrix-valued orthogonal polynomials, which can be considered as matrix-valued analogues of the Chebyshev polynomials, i.e., the spherical polynomials on (SU(2) × SU(2), diag) better known as the characters on SU(2), see [3] for the quantum group case
Summary
Matrix-valued orthogonal polynomials have been introduced and studied by Krein in connection with spectral analysis and moment problems, see [8,33,34]. We obtain a commutative algebra generated by two matrix-valued differential operators that are symmetric with respect to the matrix weight. From these differential operators, we obtain the ingredients for a matrix-valued Pearson equation which, in turn, gives us the matrix-valued adjoint of the derivative. The polynomials are eigenfunctions of the matrix-valued differential operators in Sect. We note that the matrix-valued orthogonal Gegenbauer-type polynomials of this paper are better understood in the sense of explicit formulas and for larger dimensions than the ones in [14,37,38]. For the reader’s convenience, the worksheet is available via the first author’s webpage.
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