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§18.24 Hahn Class: Asymptotic Approximations

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). 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).

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<x<\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.

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