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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: Bibliography Q
  • W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
  • W. Qiu and R. Wong (2004) Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.
  • C. K. Qu and R. Wong (1999) Best possible” upper and lower bounds for the zeros of the Bessel function J ν ( x ) . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
  • 6: 3.11 Approximation Techniques
    §3.11(i) Minimax Polynomial Approximations
    Then there exists a unique n th degree polynomial p n ( x ) , called the minimax (or best uniform) polynomial approximation to f ( x ) on [ a , b ] , that minimizes max a x b | ϵ n ( x ) | , where ϵ n ( x ) = f ( x ) - p n ( x ) . … If we have a sufficiently close approximation
    §3.11(iii) Minimax Rational Approximations
    Then the minimax (or best uniform) rational approximation
    7: 14.26 Uniform Asymptotic Expansions
    §14.26 Uniform Asymptotic Expansions
    The uniform asymptotic approximations given in §14.15 for P ν - μ ( x ) and Q ν μ ( x ) for 1 < x < are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
    8: Browsers
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    9: 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).
    10: 18.24 Hahn Class: Asymptotic Approximations
    §18.24 Hahn Class: Asymptotic Approximations
    When the parameters α and β are fixed and the ratio n / N = c is a constant in the interval(0,1), uniform asymptotic formulas (as n ) of the Hahn polynomials Q n ( z ; α , β , N ) can be found in Lin and Wong (2013) for z in three overlapping regions, which together cover the entire complex plane. … Corresponding approximations are included for the zeros of P n ( λ ) ( n x ; ϕ ) .
    Approximations in Terms of Laguerre Polynomials
    Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.