Matrix-valued Gegenbauer polynomials

We introduce matrix-valued orthogonal polynomials of arbitrary size, which are analogues of the Gegenbauer or ultraspherical polynomials for the parameter $\nu>0$. The weight function is given explicitly, and we establish positivity by proving an explicit LDU-decomposition for the weight. Several matrix-valued differential operators of order two and one are shown to be symmetric with respect to the weight, and having the matrix-valued Gegenbauer polynomials as eigenfunctions. Using the parameter $\nu$ a simple Rodrigues formula is established. The matrix-valued orthogonal polynomials are connected to the matrix-valued hypergeometric functions, which in turn allows us to give an explicit three-term recurrence relation. We give an explicit non-trivial expression for the matrix entries of the matrix-valued Gegenbauer polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using a diagonalisation procedure for a suitable matrix-valued differential operator. The case $\nu=1$ reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.