# §18.32 OP’s with Respect to Freud Weights

A Freud weight is a weight function of the form

 18.32.1 ${w(x)=\mathop{\exp\/}\nolimits\!\left(-Q(x)\right)},$ $-\infty, Symbols: $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $w(x)$: weight function and $x$: real variable Permalink: http://dlmf.nist.gov/18.32.E1 Encodings: TeX, pMML, png See also: info for 18.32

where $Q(x)$ is real, even, nonnegative, and continuously differentiable. Of special interest are the cases $Q(x)=x^{2m}$, $m=1,2,\dots$. 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 $Q(x)=x^{4}$ see Bo and Wong (1999).

For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999).