Stieltjes polynomials and Gauss-Kronrod quadrature formulae for measures induced by Chebyshev polynomials

Given a fixedn1, and a (monic) orthogonal polynomial _n (·)= _n (·;d) relative to a positive measured on the interval a, b], one can define the nonnegative measure , to which correspond the (monic) orthogonal polynomials . The coefficients in the three-term recurrence relation for , whend is a Chebyshev measure of any of the four kinds, were obtained analytically in closed form by Gautschi and Li. Here, we give explicit formulae for the Stieltjes polynomials whend is any of the four Chebyshev measures. In addition, we show that the corresponding Gauss-Kronrod quadrature formulae for each of these , based on the zeros of and , have all the desirable properties of the interlacing of nodes, their inclusion in –1, 1], and the positivity of all quadrature weights. Exceptions occur only for the Chebyshev measured of the third or fourth kind andn even, in which case the inclusion property fails. The precise degree of exactness for each of these formulae is also determined.