# best uniform polynomial approximation

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##### 1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
###### §31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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. …
##### 3: 24.1 Special Notation
###### Bernoulli Numbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
###### Euler Numbers and Polynomials
The notations $E_{n}$, $E_{n}\left(x\right)$, 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: 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 $J_{\nu}(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
• C. Quesne (2011) Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.
• ##### 6: 3.11 Approximation Techniques
###### §3.11(i) Minimax PolynomialApproximations
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 $\max_{a\leq x\leq b}\left|\epsilon_{n}(x)\right|$, where $\epsilon_{n}(x)=f(x)-p_{n}(x)$. … 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 $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ for $1 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 $n\to\infty$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)$ 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 $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to\infty$ ) of the Hahn polynomials $Q_{n}(z;\alpha,\beta,N)$ 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 $P^{(\lambda)}_{n}\left(nx;\phi\right)$.
###### Approximations in Terms of Laguerre Polynomials
Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.