18 Orthogonal PolynomialsOther Orthogonal Polynomials18.30 Associated OP’s18.32 OP’s with Respect to Freud Weights

Let $\rho (x)$ be a polynomial of degree $\mathrm{\ell}$ and positive when $-1\le x\le 1$. The Bernstein–Szegő polynomials $\{{p}_{n}(x)\}$, $n=0,1,\mathrm{\dots}$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-{x}^{2})}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-{x}^{2})}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. In consequence, ${p}_{n}(\mathrm{cos}\theta )$ can be given explicitly in terms of $\rho (\mathrm{cos}\theta )$ and sines and cosines, provided that $$ in the first case, $$ in the second case, and $$ in the third case. See Szegő (1975, §2.6).