# §18.24 Hahn Class: Asymptotic Approximations

## Hahn

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 $n\to\infty$ ) 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.

## Krawtchouk

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

## Meixner

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

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

## Charlier

Dunster (2001b) provides various asymptotic expansions for $\mathop{C_{n}\/}\nolimits\!\left(x;a\right)$ as $n\to\infty$, in terms of elementary functions or in terms of Bessel functions. Taken together, these expansions are uniformly valid for $-\infty 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).

## Meixner–Pollaczek

For an asymptotic expansion of $\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(nx;\phi\right)$ as $n\to\infty$, with $\phi$ 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 $\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(nx;\phi\right)$.

## Approximations in Terms of Laguerre Polynomials

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