§18.32 OP’s with Respect to Freud Weights

Keywords:: Freud weight function, orthogonal polynomials with Freud weights, weight functions
Referenced by:: ¶ ‣ §18.38(ii), Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/18.32
Amendment (effective with 1.0.1):: References were added at the end of this section for more general Freud weights.
Reported 2011-03-02

A Freud weight is a weight function of the form

18.32.1 ${w(x)=\mathop{\exp\/}\nolimits\!\left(-Q(x)\right)},$ $-\infty<x<\infty$ ,

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $w(x)$ : weight function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.32.E1
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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).