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1: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
For any partition κ , the zonal polynomial Z κ : 𝒮 is defined by the properties …
Normalization
Orthogonal Invariance
Summation
2: 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. …
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 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). … In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials T n ( x ) , n = 0 , 1 , , N , are orthogonal on the discrete point set comprising the zeros x N + 1 , n , n = 1 , 2 , , N + 1 , of T N + 1 ( x ) : … 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: 3.11 Approximation Techniques
§3.11(i) Minimax Polynomial Approximations
For examples of minimax polynomial approximations to elementary and special functions see Hart et al. (1968). … They satisfy the recurrence relationThe theory of polynomial minimax approximation given in §3.11(i) can be extended to the case when p n ( x ) is replaced by a rational function R k , ( x ) . …
6: 16.7 Relations to Other Functions
§16.7 Relations to Other Functions
For orthogonal polynomials see Chapter 18. …
7: 6.11 Relations to Other Functions
§6.11 Relations to Other Functions
Incomplete Gamma Function
Confluent Hypergeometric Function
6.11.2 E 1 ( z ) = e - z U ( 1 , 1 , z ) ,
8: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
9: 18.34 Bessel Polynomials
§18.34 Bessel Polynomials
§18.34(i) Definitions and Recurrence Relation
Because the coefficients C n in (18.34.4) are not all positive, the polynomials y n ( x ; a ) cannot be orthogonal on the line with respect to a positive weight function. … For uniform asymptotic expansions of y n ( x ; a ) as n in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c). …
10: 13.18 Relations to Other Functions
§13.18 Relations to Other Functions
§13.18(iv) Parabolic Cylinder Functions
§13.18(v) Orthogonal Polynomials
Hermite Polynomials
Laguerre Polynomials