When the parameters and are fixed and the ratio is a constant in the interval(0,1), uniform asymptotic formulas (as ) of the Hahn polynomials can be found in Lin and Wong (2013) for in three overlapping regions, which together cover the entire complex plane. In particular, asymptotic formulas in terms of elementary functions are given when is real and fixed.
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 .
For two asymptotic expansions of as , with and fixed, see Jin and Wong (1998) and Wang and Wong (2011). 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.
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).
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 .