18 Orthogonal PolynomialsOther Orthogonal Polynomials18.31 Bernstein–Szegö Polynomials18.33 Polynomials Orthogonal on the Unit Circle

A *Freud weight* is a weight function of the form

18.32.1 | $$w\left(x\right)=\mathrm{exp}\left(-Q\left(x\right)\right),$$ | ||

$$, | |||

where $Q\left(x\right)$ is real, even, nonnegative, and continuously differentiable. Of special interest are the cases $Q\left(x\right)={x}^{2m}$, $m=1,2,\mathrm{\dots}$. No explicit expressions for the corresponding OP’s are available. However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q\left(x\right)={x}^{4}$ see Bo and Wong (1999).