18 Orthogonal PolynomialsAskey Scheme18.23 Hahn Class: Generating Functions18.25 Wilson Class: Definitions

With $x=\lambda N$ and $\nu =n/N$, Li and Wong (2000) gives an asymptotic expansion for ${K}_{n}\left(x;p,N\right)$ as $n\to \mathrm{\infty}$, that holds uniformly for $\lambda $ and $\nu $ in compact subintervals of $\left(0,1\right)$. This expansion is in terms of the parabolic cylinder function and its derivative.

With $\mu =N/n$ and $x$ fixed, Qiu and Wong (2004) gives an asymptotic expansion for ${K}_{n}\left(x;p,N\right)$ as $n\to \mathrm{\infty}$, that holds uniformly for $\mu \in \left[1,\mathrm{\infty}\right)$. This expansion is in terms of confluent hypergeometric functions. Asymptotic approximations are also provided for the zeros of ${K}_{n}\left(x;p,N\right)$ in various cases depending on the values of $p$ and $\mu $.

For two asymptotic expansions of ${M}_{n}\left(nx;\beta ,c\right)$ as $n\to \mathrm{\infty}$, with $\beta $ and $c$ fixed, see Jin and Wong (1998) and Wang and Wong (2011). The first expansion holds uniformly for $\delta \le x\le 1+\delta $, and the second for $1-\delta \le x\le 1+{\delta}^{-1}$, $\delta $ being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions.

Dunster (2001b) provides various asymptotic expansions for ${C}_{n}\left(x,a\right)$ as $n\to \mathrm{\infty}$, in terms of elementary functions or in terms of Bessel functions. Taken together, these expansions are uniformly valid for $$ and for $a$ in unbounded intervals—each of which contains $\left[0,\left(1-\delta \right)n\right]$, where $\delta $ again denotes an arbitrary small positive constant. See also Bo and Wong (1994) and Goh (1998).

For an asymptotic expansion of ${P}_{n}^{\left(\lambda \right)}\left(nx;\varphi \right)$ as $n\to \mathrm{\infty}$, with $\varphi $ fixed, see Li and Wong (2001). This expansion is uniformly valid in any compact $x$-interval on the real line and is in terms of parabolic cylinder functions. Corresponding approximations are included for the zeros of ${P}_{n}^{\left(\lambda \right)}\left(nx;\varphi \right)$.

For asymptotic approximations to ${P}_{n}^{\left(\lambda \right)}\left(x;\varphi \right)$ as $\left|x+\mathrm{i}\lambda \right|\to \mathrm{\infty}$, with $n$ fixed, see Temme and López (2001). These approximations are in terms of Laguerre polynomials and hold uniformly for $\mathrm{ph}\left(x+\mathrm{i}\lambda \right)\in \left[0,\pi \right]$. Compare also (18.21.12). Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.