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continuous q-ultraspherical polynomials

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.2 Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) = H ( a , q n , m ; - n , β , γ , δ ; z )
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . These solutions are the Heun polynomials. …
2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
Normalization
Orthogonal Invariance
Summation
Mean-Value
3: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
4: 18.3 Definitions
§18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , see Abramowitz and Stegun (1964, pp. 793–801). … For another version of the discrete orthogonality property of the polynomials T n ( x ) see (3.11.9). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
5: 18.28 Askey–Wilson Class
§18.28(v) Continuous q -Ultraspherical Polynomials
These polynomials are also called Rogers polynomials.
§18.28(vi) Continuous q -Hermite Polynomials
§18.28(vii) Continuous q - 1 -Hermite Polynomials
For continuous q - 1 -Hermite polynomials the orthogonality measure is not unique. …
6: 18.30 Associated OP’s
§18.30 Associated OP’s
In the recurrence relation (18.2.8) assume that the coefficients A n , B n , and C n + 1 are defined when n is a continuous nonnegative real variable, and let c be an arbitrary positive constant. …
Associated Jacobi Polynomials
Associated Legendre Polynomials
7: 18.1 Notation
  • Continuous Hahn: p n ( x ; a , b , a ¯ , b ¯ ) .

  • Continuous Dual Hahn: S n ( x ; a , b , c ) .

  • Continuous q -Ultraspherical: C n ( x ; β | q ) .

  • Continuous q -Hermite: H n ( x | q ) .

  • Continuous q - 1 -Hermite: h n ( x | q )

  • 8: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
    §18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
    Ismail (1986) gives asymptotic expansions as n , with x and other parameters fixed, for continuous q -ultraspherical, big and little q -Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson p n ( cos θ ; a , b , c , d | q ) the leading term is given by … For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). For asymptotic approximations to the largest zeros of the q -Laguerre and continuous q - 1 -Hermite polynomials see Chen and Ismail (1998).
    9: 18.21 Hahn Class: Interrelations
    §18.21 Hahn Class: Interrelations
    §18.21(i) Dualities
    §18.21(ii) Limit Relations and Special Cases
    Hahn Jacobi
    Continuous Hahn Meixner–Pollaczek
    10: 18.26 Wilson Class: Continued
    Wilson Continuous Dual Hahn
    Wilson Continuous Hahn
    Continuous Dual Hahn Meixner–Pollaczek
    Continuous Dual Hahn
    Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …