A set of orthogonal polynomials induced by a given orthogonal polynomial

Summary

Given an integern ⩾ 1, and the orthogonal polynomialsπ _n (·; dσ) of degreen relative to some positive measuredσ, the polynomial system “induced” byπ _n is the system of orthogonal polynomials$\{ \hat \pi _{k,n} \}$ corresponding to the modified measure$d\hat \sigma _n = \pi _n^2 d\sigma$. Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials$\hat \pi _{k,n}$ from the recursion coefficients of the orthogonal polynomials belonging to the measuredσ. A stable computational algorithm is proposed, which uses a sequence ofQR steps with shifts. For all four Chebyshev measuresdσ, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by Al-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros ofπ _n and those of$\hat \pi _{n + 1,n}$ are studied for Gegenbauer measures, as well as the orthogonality—or lack thereof—of the polynomial sequence$\{ \hat \pi _{n,n - 1} \}$.