# 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

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31.5.2
$${\mathit{Hp}}_{n,m}(a,{q}_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell}(a,{q}_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

►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

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###### 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

►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 ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials ${C}_{n}^{(\lambda )}\left(x\right)$, the Chebyshev polynomials ${T}_{n}\left(x\right)$ and ${U}_{n}\left(x\right)$, the Legendre polynomials ${P}_{n}\left(x\right)$, the Laguerre polynomials ${L}_{n}\left(x\right)$, and the Hermite polynomials ${H}_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … ►For another version of the discrete orthogonality property of the polynomials ${T}_{n}\left(x\right)$ see (3.11.9). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …##### 5: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points.
Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
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Asymptotic expansion of the Krawtchouk polynomials and their zeros.
Comput. Methods Funct. Theory 4 (1), pp. 189–226.
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“Best possible” upper and lower bounds for the zeros of the Bessel function ${J}_{\nu}(x)$
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Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

##### 6: 3.11 Approximation Techniques

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###### §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 ${\mathrm{max}}_{a\le x\le b}\left|{\u03f5}_{n}(x)\right|$, where ${\u03f5}_{n}(x)=f(x)-{p}_{n}(x)$. … ►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 ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(x\right)$ for $$ 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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