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quadratic Jacobi symbol

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1: 27.9 Quadratic Characters
If an odd integer P has prime factorization P = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( n | p r ) a r , with ( n | 1 ) = 1 . …
2: Bibliography
  • R. Askey and J. Fitch (1969) Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.
  • R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.
  • R. Askey and B. Razban (1972) An integral for Jacobi polynomials. Simon Stevin 46, pp. 165–169.
  • R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
  • M. J. Atia, A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet (2014) Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.
  • 3: 18.7 Interrelations and Limit Relations
    Ultraspherical and Jacobi
    Chebyshev, Ultraspherical, and Jacobi
    §18.7(ii) Quadratic Transformations
    Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). …
    Jacobi Hermite
    4: 18.19 Hahn Class: Definitions
    The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator d d x in the case of the classical OP’s is played by a suitable difference operator. …
  • 2.

    Wilson class (or quadratic lattice class). These are OP’s p n ( x ) = p n ( λ ( y ) ) ( p n ( x ) of degree n in x , λ ( y ) quadratic in y ) where the role of the differentiation operator is played by Δ y Δ y ( λ ( y ) ) or y y ( λ ( y ) ) or δ y δ y ( λ ( y ) ) . The Wilson class consists of two discrete and two continuous families.

  • 18.19.5 k n = ( n + 2 ( a + b ) 1 ) n n ! .
    5: 18.33 Polynomials Orthogonal on the Unit Circle
    After a quadratic transformation (18.2.23) this would express OP’s on [ 1 , 1 ] with an even orthogonality measure in terms of the ϕ n . …
    18.33.13 ϕ n ( z ) = = 0 n ( λ + 1 ) ( λ ) n ! ( n ) ! z = ( λ ) n n ! F 1 2 ( n , λ + 1 λ n + 1 ; z ) ,
    Askey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle. …
    18.33.15 ϕ n ( z ) = = 0 n ( a q 2 ; q 2 ) ( a ; q 2 ) n ( q 2 ; q 2 ) ( q 2 ; q 2 ) n ( q 1 z ) = ( a ; q 2 ) n ( q 2 ; q 2 ) n ϕ 1 2 ( a q 2 , q 2 n a 1 q 2 2 n ; q 2 , q z a ) ,
    18.33.16 w ( z ) = | ( q z ; q 2 ) / ( a q z ; q 2 ) | 2 , a 2 q 2 < 1 .
    6: 18.27 q -Hahn Class
    The q -hypergeometric OP’s comprise the q -Hahn class (or q -linear lattice class) OP’s and the Askey–Wilson class (or q -quadratic lattice class) OP’s (§18.28). …
    §18.27(iii) Big q -Jacobi Polynomials
    From Big q -Jacobi to Jacobi
    From Big q -Jacobi to Little q -Jacobi
    From Little q -Jacobi to Jacobi