Jacobi polynomials from compatibility conditions

Abstract

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)(alpha) (1+x)beta, x is an element of [-1, 1], we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials.