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18 Orthogonal PolynomialsAskey Scheme

§18.24 Hahn Class: Asymptotic Approximations

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.