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§18.24 Hahn Class: Asymptotic Approximations

Hahn

When the parameters α and β are fixed and the ratio n/N=c is a constant in the interval(0,1), uniform asymptotic formulas (as n ) of the Hahn polynomials Qn(z;α,β,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=λN and ν=n/N, Li and Wong (2000) gives an asymptotic expansion for Kn(x;p,N) as n, that holds uniformly for λ and ν in compact subintervals of (0,1). This expansion is in terms of the parabolic cylinder function and its derivative.

With μ=N/n and x fixed, Qiu and Wong (2004) gives an asymptotic expansion for Kn(x;p,N) as n, that holds uniformly for μ[1,). This expansion is in terms of confluent hypergeometric functions. Asymptotic approximations are also provided for the zeros of Kn(x;p,N) in various cases depending on the values of p and μ.

Meixner

For two asymptotic expansions of Mn(nx;β,c) as n, with β and c fixed, see Jin and Wong (1998) and Wang and Wong (2011). The first expansion holds uniformly for δx1+δ, and the second for 1-δx1+δ-1, δ being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions.

For asymptotic approximations for the zeros of Mn(nx;β,c) in terms of zeros of Ai(x)9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012).

Charlier

Dunster (2001b) provides various asymptotic expansions for Cn(x;a) as n, in terms of elementary functions or in terms of Bessel functions. Taken together, these expansions are uniformly valid for -<x< and for a in unbounded intervals—each of which contains [0,(1-δ)n], where δ again denotes an arbitrary small positive constant. See also Bo and Wong (1994) and Goh (1998).

Meixner–Pollaczek

For an asymptotic expansion of Pn(λ)(nx;ϕ) as n, with ϕ 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 Pn(λ)(nx;ϕ).

Approximations in Terms of Laguerre Polynomials

For asymptotic approximations to Pn(λ)(x;ϕ) as |x+iλ|, with n fixed, see Temme and López (2001). These approximations are in terms of Laguerre polynomials and hold uniformly for ph(x+iλ)[0,π]. Compare also (18.21.12). Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.