§18.24 Hahn Class: Asymptotic Approximations

When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. In particular, asymptotic formulas in terms of elementary functions are given when $z=x$ is real and fixed.

With $x=\lambda N$ and $\nu =n/N$, Li and Wong (2000) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\lambda$ and $\nu$ in compact subintervals of $(0,1)$. 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 $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\mu \in [1,\mathrm{\infty })$. This expansion is in terms of confluent hypergeometric functions. Asymptotic approximations are also provided for the zeros of $K_n(x;p,N)$ in various cases depending on the values of $p$ and $\mu$.

For two asymptotic expansions of $M_n(nx;\beta ,c)$ 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.

For asymptotic approximations for the zeros of $M_n(nx;\beta ,c)$ in terms of zeros of $\mathrm{Ai}\left(x\right)$ (§9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012).

Dunster (2001b) provides various asymptotic expansions for $C_n(x;a)$ 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 $[0,(1-\delta )n]$, where $\delta$ again denotes an arbitrary small positive constant. See also Bo and Wong (1994) and Goh (1998).

For an asymptotic expansion of $P_n^{(\lambda )}(nx;\varphi )$ 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 $P_n^{(\lambda )}(nx;\varphi )$.

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