Recurrence relations of the multi-indexed orthogonal polynomials. III

In Paper II, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which gives a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gomez-Ullate et al. [J. Approx. Theory 204, 1 (2016); e-print arXiv: 1506.03651 [math. CA]] to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of Hermite type. (C) 2016 AIP Publishing LLC.