A Set of Orthogonal Polynomials Induced by a Given Orthogonal Polynomial

Given an integer n ~ 1, and the orthogonal polynomial1rn (· ; du) of degree n relative to some positive measure du, the polynomial system "induced" by 1rn is the system of orthogonal polynomials {7rk,n} corresponding to the modified measure dUn = 1r~du. Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials 7rk,n from the recursion coefficients of the orthogonal polynomials belonging to the measure du. A stable computational algorithm is proposed, which uses a sequence of QR steps with shifts. For all four Chebyshev measures du, the desired coefficients are obtained analytically in closed form, which in part reproduces (with different methods) results obtained previously by AI-Salam, Allaway and Askey via sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Interlacing properties involving the zeros of 1rn and those of 7rn+l,n are studied for Gegenbauer measures, as well as the orthogonality or lack thereof of the polynomial sequence {7rn,n-Il·