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Meixner–Pollaczek polynomials

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1: 18.24 Hahn Class: Asymptotic Approximations
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
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 ) .
MeixnerPollaczek Laguerre
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 ) ,
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
18.19.6 p n ( x ) = P n ( λ ) ( x ; ϕ ) ,
5: 18.20 Hahn Class: Explicit Representations
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
18.23.7 ( 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 ) .
  • MeixnerPollaczek: 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 ) .
    10: 15.9 Relations to Other Functions