Jacobi Polynomials From Compatibility Conditions

We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable z (spectral parameter) and the other a recurrence relation in n (the lattice variable). For the Jacobi weight w(x) = (1-x) α(1 + x) β, x ∈ [-1, 1], we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials