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1: 18.24 Hahn Class: Asymptotic Approximations
With x = λ N and ν = n / N , Li and Wong (2000) gives an asymptotic expansion for K n ( x ; p , N ) as n , that holds uniformly for λ and ν in compact subintervals of ( 0 , 1 ) . … With μ = N / n and x fixed, Qiu and Wong (2004) gives an asymptotic expansion for K n ( x ; p , N ) as n , that holds uniformly for μ [ 1 , ) . …Asymptotic approximations are also provided for the zeros of K n ( x ; p , N ) in various cases depending on the values of p and μ . … Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.
2: 18.21 Hahn Class: Interrelations
K n ( x ; p , N ) = K x ( n ; p , N ) , n , x = 0 , 1 , , N .
18.21.3 lim t Q n ( x ; p t , ( 1 - p ) t , N ) = K n ( x ; p , N ) .
18.21.6 lim N K n ( x ; N - 1 a , N ) = C n ( x ; a ) .
3: 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
K n ( x ; p , N ) , n = 0 , 1 , , N { 0 , 1 , , N } ( N x ) p x ( 1 - p ) N - x , 0 < p < 1 ( 1 - p p ) n / ( N n )
Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.
p n ( x ) k n
K n ( x ; p , N ) p - n / ( - N ) n
4: 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
K n ( x ; p , N ) p ( N - n ) n ( 1 - p )
Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.
p n ( x ) A ( x ) C ( x ) λ n
K n ( x ; p , N ) p ( x - N ) ( p - 1 ) x - n
18.22.21 Δ x K n ( x ; p , N ) = - n p N K n - 1 ( x ; p , N - 1 ) ,
18.22.22 x ( ( N x ) p x ( 1 - p ) N - x K n ( x ; p , N ) ) = ( N + 1 x ) p x ( 1 - p ) N - x K n + 1 ( x ; p , N + 1 ) .
5: 18.38 Mathematical Applications
Coding Theory
For applications of Krawtchouk polynomials K n ( x ; p , N ) and q -Racah polynomials R n ( x ; α , β , γ , δ | q ) to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).
6: 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
K n ( x ; p , N ) x - N ( - N ) n
7: Bibliography Q
  • W. Qiu and R. Wong (2004) Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.
  • 8: 18.23 Hahn Class: Generating Functions
    18.23.3 ( 1 - 1 - p p z ) x ( 1 + z ) N - x = n = 0 N ( N n ) K n ( x ; p , N ) z n , x = 0 , 1 , , N .
    9: 18.1 Notation
    ( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
  • Krawtchouk: K n ( x ; p , N ) .

  • 10: 18.26 Wilson Class: Continued
    18.26.11 lim t R n ( x ( x + t + 1 ) ; p t , ( 1 - p ) t , N ) = K n ( x ; p , N ) .