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1: 18.24 Hahn Class: Asymptotic Approximations
For two asymptotic expansions of M n ( n x ; β , c ) as n , with β and c fixed, see Jin and Wong (1998) and Wang and Wong (2011). … For asymptotic approximations for the zeros of M n ( n x ; β , c ) in terms of zeros of Ai ( x ) 9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … For an asymptotic expansion of P n ( λ ) ( n x ; ϕ ) as n , with ϕ fixed, see Li and Wong (2001). …Corresponding approximations are included for the zeros of P n ( λ ) ( n x ; ϕ ) . … For asymptotic approximations to P n ( λ ) ( x ; ϕ ) as | x + i λ | , with n fixed, see Temme and López (2001). …
2: 18.21 Hahn Class: Interrelations
M n ( x ; β , c ) = M x ( n ; β , c ) , n , x = 0 , 1 , 2 , .
18.21.7 lim β M n ( x ; β , a ( a + β ) - 1 ) = C n ( x ; a ) .
18.21.10 lim t t - n p n ( x - t ; λ + i t , - t tan ϕ , λ - i t , - t tan ϕ ) = ( - 1 ) n ( cos ϕ ) n P n ( λ ) ( x ; ϕ ) .
18.21.12 lim ϕ 0 P n ( 1 2 α + 1 2 ) ( - ( 2 ϕ ) - 1 x ; ϕ ) = L n ( α ) ( x ) .
3: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.7 p n ( x ) = P n ( λ ) ( x ; ϕ ) ,
Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.
p n ( x ) A ( x ) C ( x ) λ n
M n ( x ; β , c ) c ( x + β ) x n ( 1 - c )
18.22.16 p n ( x ) = P n ( λ ) ( x ; ϕ ) ,
18.22.29 δ x ( P n ( λ ) ( x ; ϕ ) ) = 2 sin ϕ P n - 1 ( λ + 1 2 ) ( x ; ϕ ) ,
18.22.30 δ x ( w ( λ + 1 2 ) ( x ; ϕ ) P n ( λ + 1 2 ) ( x ; ϕ ) ) = - ( n + 1 ) w ( λ ) ( x ; ϕ ) P n + 1 ( λ ) ( x ; ϕ ) .
4: 18.19 Hahn Class: Definitions
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
M n ( x ; β , c ) { 0 , 1 , 2 , } ( β ) x c x / x ! , β > 0 , 0 < c < 1 c - n n ! ( β ) n ( 1 - c ) β
Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.
p n ( x ) k n
M n ( x ; β , c ) ( 1 - c - 1 ) n / ( β ) n
18.19.6 p n ( x ) = P n ( λ ) ( x ; ϕ ) ,
5: 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
M n ( x ; β , c ) x + β ( β ) n
18.20.4 w ( λ ) ( x ; ϕ ) P n ( λ ) ( x ; ϕ ) = 1 n ! δ x n ( w ( λ + 1 2 n ) ( x ; ϕ ) ) .
6: 18.23 Hahn Class: Generating Functions
18.23.4 ( 1 - z c ) x ( 1 - z ) - x - β = n = 0 ( β ) n n ! M n ( x ; β , c ) z n , x = 0 , 1 , 2 , , | z | < 1 .
18.23.7 ( 1 - e i ϕ z ) - λ + i x ( 1 - e - i ϕ z ) - λ - i x = n = 0 P n ( λ ) ( x ; ϕ ) z n , | z | < 1 .
7: 18.35 Pollaczek Polynomials
For the polynomials C n ( λ ) ( x ) and P n ( λ ) ( x ; ϕ ) see §§18.3 and 18.19, respectively. …
8: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
  • Meixner: M n ( x ; β , c ) .

  • Meixner–Pollaczek: P n ( λ ) ( x ; ϕ ) .

  • 9: 18.26 Wilson Class: Continued
    18.26.8 lim t S n ( ( x - t ) 2 ; λ + i t , λ - i t , t cot ϕ ) / t n = n ! ( csc ϕ ) n P n ( λ ) ( x ; ϕ ) .
    18.26.13 lim N R n ( r ( x ; β , c , N ) ; β - 1 , c - 1 ( 1 - c ) N , N ) = M n ( x ; β , c ) .
    10: 15.9 Relations to Other Functions