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continuous q-Hermite 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
For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to n = 12 for these polynomials and for coefficients up to n = 6 for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …
Bessel polynomials
Bessel polynomials are often included among the classical OP’s. …
5: 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 )

  • 6: 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).
    7: 18.25 Wilson Class: Definitions
    Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) . … Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is ( 0 , ) S , where S is a specific finite set, e. …
    §18.25(ii) Weights and Standardizations: Continuous Cases
    Continuous Dual Hahn
    Table 18.25.2 provides the leading coefficients k n 18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …
    8: 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
    9: 18.19 Hahn Class: Definitions
    §18.19 Hahn Class: Definitions
  • 1.

    Hahn class (or linear lattice class). These are OP’s p n ( x ) where the role of d d x is played by Δ x or x or δ x (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

  • The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). …
    Continuous Hahn
    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. …