# §18.24 Hahn Class: Asymptotic Approximations

## ¶ Krawtchouk

With and , Li and Wong (2000) gives an asymptotic expansion for as , that holds uniformly for and in compact subintervals of . This expansion is in terms of the parabolic cylinder function and its derivative.

With and fixed, Qiu and Wong (2004) gives an asymptotic expansion for as , that holds uniformly for . This expansion is in terms of confluent hypergeometric functions. Asymptotic approximations are also provided for the zeros of in various cases depending on the values of and .

## ¶ Meixner

For two asymptotic expansions of as , with and fixed, see Jin and Wong (1998). The first expansion holds uniformly for , and the second for , being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions.

For asymptotic approximations for the zeros of in terms of zeros of 9.9(i)), see Jin and Wong (1999).

## ¶ Charlier

Dunster (2001b) provides various asymptotic expansions for as , in terms of elementary functions or in terms of Bessel functions. Taken together, these expansions are uniformly valid for and for in unbounded intervals—each of which contains , where again denotes an arbitrary small positive constant. See also Bo and Wong (1994) and Goh (1998).

## ¶ Meixner–Pollaczek

For an asymptotic expansion of as , with fixed, see Li and Wong (2001). This expansion is uniformly valid in any compact -interval on the real line and is in terms of parabolic cylinder functions. Corresponding approximations are included for the zeros of .

## ¶ Approximations in Terms of Laguerre Polynomials

For asymptotic approximations to as , with fixed, see Temme and López (2001). These approximations are in terms of Laguerre polynomials and hold uniformly for . Compare also (18.21.12). Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.