Journal of Lie Theory

\def\a{{\frak a}} \def\g{{\frak g}} \def\k{{\frak k}} We obtain the LU-decomposition of a non commutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\g)^{K}\to U(\k)^{M} \otimes U(\a)$. Although this system can not be expressed as a single matrix equation with coefficients in $U(\k)$, it turns out that obtaining a triangular system equivalent to it, can be reduced to obtaining the LU-decomposition of a matrix $\widetilde M_0$ with entries in a polynomial algebra. We prove that both the L-part and U-part of $\widetilde M_0$ are expressed in terms of Jacobi polynomials. Moreover, each entry of the L-part of $\widetilde M_0$ and of its inverse is given by a single ultraspherical Jacobi polynomial. This fact yields a biorthogonality relation between the ultraspherical Jacobi polynomials.