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1: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … Whittaker’s notation D ν ( z ) is useful when ν is a nonnegative integer (Hermite polynomial case).
2: 12.7 Relations to Other Functions
§12.7(i) Hermite Polynomials
3: 12.11 Zeros
Lastly, when a = - n - 1 2 , n = 1 , 2 , (Hermite polynomial case) U ( a , x ) has n zeros and they lie in the interval [ - 2 - a , 2 - a ] . …
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)). This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … 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). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
5: 13.18 Relations to Other Functions
Special cases are the error functions …
§13.18(v) Orthogonal Polynomials
Special cases of §13.18(iv) are as follows. …
Hermite Polynomials
Laguerre Polynomials
6: 13.6 Relations to Other Functions
Special cases are the error functions … and in the case that b - 2 a is an integer we have …Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively … Special cases of §13.6(iv) are as follows. …
Hermite Polynomials
7: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1.
Laguerre
Table 18.6.1: Classical OP’s: symmetry and special values.
p n ( x ) p n ( - x ) p n ( 1 ) p 2 n ( 0 ) p 2 n + 1 ( 0 )
§18.6(ii) Limits to Monomials
18.6.4 lim λ C n ( λ ) ( x ) C n ( λ ) ( 1 ) = x n ,
8: 18.15 Asymptotic Approximations
§18.15(i) Jacobi
§18.15(v) Hermite
With μ = 2 n + 1 the expansions in Chapter 12 are for the parabolic cylinder function U ( - 1 2 μ 2 , μ t 2 ) , which is related to the Hermite polynomials via … For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). These approximations apply when the parameters are large, namely α and β (subject to restrictions) in the case of Jacobi polynomials, λ in the case of ultraspherical polynomials, and | α | + | x | in the case of Laguerre polynomials. …
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
Charlier Hermite
10: 18.18 Sums
Hermite
Hermite
Hermite
Hermite
Hermite