§18.32 OP’s with Respect to Freud Weights

A Freud weight is a weight function of the form

where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime }(x)$ increases for $x>0$, and $Q^{\prime }(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, see Freud (1969). These conditions on $Q(x)$ have been strengthened and also relaxed in literature. See the early survey by Nevai (1986, Part 2). Of special interest are the cases $Q(x)=x^{2m}$, $m=1,2,\mathrm{\dots }$, and the case $Q(x)=\frac{1}{4}{x}^4-t{x}^2$ ($t\in ℝ$), see §32.15. 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).

Generalized Freud weights have the form

with $Q(x)$ satisfying similar conditions as before, see Kasuga and Sakai (2003). The case $Q(x)={\left|x\right|}^{\beta }$ ($\beta >0$) was already introduced by Freud (1976). The special case $Q(x)=\frac{1}{4}{x}^4-t{x}^2$ is of particular interest, see Clarkson and Jordaan (2018).

All of these forms appear in applications, see §18.39(iii) and Table 18.39.1, albeit sometimes with $x\in [0,\mathrm{\infty })$, where the term half-Freud weight is used; or on $x\in [-1,1]$ or $[0,1]$, where the term Rys weight is employed, see Rys et al. (1983). For (generalized) Freud weights on a subinterval of $[0,\mathrm{\infty })$ see also Levin and Lubinsky (2005).