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 z (spectral parameter) and the other a recurrence relation in n n (the lattice variable). For the Jacobi weight \[ w ( x ) = ( 1 − x ) α ( 1 + x ) β , x ∈ [ − 1 , 1 ] , w(x)=(1-x)^{\alpha }(1+x)^{\beta },\qquad x\in [-1,1], \] we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials.