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1: 18.21 Hahn Class: Interrelations
C n ( x ; a ) = C x ( n ; a ) , n , x = 0 , 1 , 2 , .
18.21.6 lim N K n ( x ; N - 1 a , N ) = C n ( x ; a ) .
18.21.7 lim β M n ( x ; β , a ( a + β ) - 1 ) = C n ( x ; a ) .
Charlier Hermite
18.21.9 lim a ( 2 a ) 1 2 n C n ( ( 2 a ) 1 2 x + a ; a ) = ( - 1 ) n H n ( x ) .
2: 18.19 Hahn Class: Definitions
Hahn, Krawtchouk, Meixner, and Charlier
Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§18.2(i), 18.2(iii)) for the Hahn polynomials Q n ( x ; α , β , N ) , Krawtchouk polynomials K n ( x ; p , N ) , Meixner polynomials M n ( x ; β , c ) , and Charlier polynomials C n ( x ; a ) .
Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.
p n ( x ) X w x h n
C n ( x ; a ) { 0 , 1 , 2 , } a x / x ! , a > 0 a - n e a n !
Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.
p n ( x ) k n
C n ( x ; a ) ( - a ) - n
3: 18.22 Hahn Class: Recurrence Relations and Differences
Table 18.22.1: Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials.
p n ( x ) A n C n
C n ( x ; a ) a n
Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.
p n ( x ) A ( x ) C ( x ) λ n
C n ( x ; a ) a x n
18.22.25 Δ x C n ( x ; a ) = - n a C n - 1 ( x ; a ) ,
18.22.26 x ( a x x ! C n ( x ; a ) ) = a x x ! C n + 1 ( x ; a ) .
4: 18.20 Hahn Class: Explicit Representations
For the Krawtchouk, Meixner, and Charlier polynomials, F ( x ) and κ n are as in Table 18.20.1.
Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).
p n ( x ) F ( x ) κ n
C n ( x ; a ) 1 1
5: 18.23 Hahn Class: Generating Functions
18.23.5 e z ( 1 - z a ) x = n = 0 C n ( x ; a ) n ! z n , x = 0 , 1 , 2 , .
6: 18.24 Hahn Class: Asymptotic Approximations
Dunster (2001b) provides various asymptotic expansions for C n ( x ; a ) as n , in terms of elementary functions or in terms of Bessel functions. …
7: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
  • Charlier: C n ( x ; a ) .

  • 8: 13.6 Relations to Other Functions
    9: Bibliography D
  • T. M. Dunster (2001b) Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.
  • 10: Bibliography G
  • W. M. Y. Goh (1998) Plancherel-Rotach asymptotics for the Charlier polynomials. Constr. Approx. 14 (2), pp. 151–168.