Recurrence Coefficients of the Orthogonal Polynomials for Oscillatory Jacobi-type Weight Functions

Abstract

We study two classes of oscillatory Jacobi-type weight functions $x^c(1-x^2)^{λ-1/2}\exp(iζx)$, $x\in[-1,1], λ>-1/2, c\in\{0,1\}$. Other restrictions are imposed on $λ$ and $ζ$ to guarantee the existence of the associated orthogonal polynomials. By using the ladder operators established in the recent literature for monic orthogonal polynomials associated with Jacobi-type weight functions and three compatibility conditions, we derive two coupled difference equations satisfied by the three-term recurrence coefficients. Compared with the existing results, these equations are structurally simpler and of lower order. Once the initial values are determined, the recurrence coefficients can be computed at any stage through the difference equations. The obtained expressions enable us to conjecture the symbolic forms for the recurrence coefficients.