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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: 18.19 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
    6: 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 )

  • 7: 18.25 Wilson Class: Definitions
    The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn). 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 ) . …
    §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: 18.26 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. …
    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: 18.29 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).