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Askey?Wilson polynomials

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1: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
For Askey–Wilson p n ( cos θ ; a , b , c , d | q ) the leading term is given by …
2: 18.28 Askey–Wilson Class
18.28.1 p n ( x ) = p n ( x ; a , b , c , d | q ) = a n = 0 n q ( a b q , a c q , a d q ; q ) n ( q n , a b c d q n 1 ; q ) ( q ; q ) j = 0 1 ( 1 2 a q j x + a 2 q 2 j ) ,
18.28.1_5 R n ( z ) = R n ( z ; a , b , c , d | q ) = p n ( 1 2 ( z + z 1 ) ; a , b , c , d | q ) a n ( a b , a c , a d ; q ) n = ϕ 3 4 ( q n , a b c d q n 1 , a z , a z 1 a b , a c , a d ; q , q ) .
The polynomials p n ( x ; a , b , c , d | q ) are symmetric in the parameters a , b , c , d . …
18.28.7 Q n ( cos θ ; a , b | q ) = p n ( cos θ ; a , b , 0 , 0 | q ) = a n = 0 n q ( a b q ; q ) n ( q n ; q ) ( q ; q ) j = 0 1 ( 1 2 a q j cos θ + a 2 q 2 j ) = ( a b ; q ) n a n ϕ 2 3 ( q n , a e i θ , a e i θ a b , 0 ; q , q ) = ( b e i θ ; q ) n e i n θ ϕ 1 2 ( q n , a e i θ b 1 q 1 n e i θ ; q , b 1 q e i θ ) .
18.28.29 lim q 1 p n ( 1 1 2 x ( 1 q ) 2 ; q a , q b , q c , q d | q ) ( 1 q ) 3 n = W n ( x ; a , b , c , d ) .
3: 18.1 Notation
  • Askey–Wilson: p n ( x ; a , b , c , d | q ) .

  • 4: Errata
  • Equation (18.28.1)
    18.28.1 p n ( x ) = p n ( x ; a , b , c , d | q ) = a n = 0 n q ( a b q , a c q , a d q ; q ) n ( q n , a b c d q n 1 ; q ) ( q ; q ) j = 0 1 ( 1 2 a q j x + a 2 q 2 j ) ,
    18.28.1_5 R n ( z ) = R n ( z ; a , b , c , d | q ) = p n ( 1 2 ( z + z 1 ) ; a , b , c , d | q ) a n ( a b , a c , a d ; q ) n = ϕ 3 4 ( q n , a b c d q n 1 , a z , a z 1 a b , a c , a d ; q , q )

    Previously we presented all the information of these formulas in one equation

    p n ( cos θ ) = p n ( cos θ ; a , b , c , d | q ) = a n = 0 n q ( a b q , a c q , a d q ; q ) n ( q n , a b c d q n 1 ; q ) ( q ; q ) j = 0 1 ( 1 2 a q j cos θ + a 2 q 2 j ) = a n ( a b , a c , a d ; q ) n ϕ 3 4 ( q n , a b c d q n 1 , a e i θ , a e i θ a b , a c , a d ; q , q ) .