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1: 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)). …
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
Hermite H n ( x ) ( - , ) e - x 2 π 1 2 2 n n ! 2 n 0
Hermite He n ( x ) ( - , ) e - 1 2 x 2 ( 2 π ) 1 2 n ! 1 0
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). …
2: 18.41 Tables
For P n ( x ) ( = P n ( x ) ) see §14.33. Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates T n ( x ) , U n ( x ) , L n ( x ) , and H n ( x ) for n = 0 ( 1 ) 12 . The ranges of x are 0.2 ( .2 ) 1 for T n ( x ) and U n ( x ) , and 0.5 , 1 , 3 , 5 , 10 for L n ( x ) and H n ( x ) . The precision is 10D, except for H n ( x ) which is 6-11S. … For P n ( x ) , L n ( x ) , and H n ( x ) see §3.5(v). …
3: 7.10 Derivatives
§7.10 Derivatives
7.10.1 d n + 1 erf z d z n + 1 = ( - 1 ) n 2 π H n ( z ) e - z 2 , n = 0 , 1 , 2 , .
For the Hermite polynomial H n ( z ) see §18.3. …
4: 28.9 Zeros
For q the zeros of ce 2 n ( z , q ) and se 2 n + 1 ( z , q ) approach asymptotically the zeros of He 2 n ( q 1 / 4 ( π - 2 z ) ) , and the zeros of ce 2 n + 1 ( z , q ) and se 2 n + 2 ( z , q ) approach asymptotically the zeros of He 2 n + 1 ( q 1 / 4 ( π - 2 z ) ) . Here He n ( z ) denotes the Hermite polynomial of degree n 18.3). …
5: 18.38 Mathematical Applications
Integrable Systems
18.38.1 V n ( x ) = 2 n H n + 1 ( x ) H n - 1 ( x ) / ( H n ( x ) ) 2 ,
with H n ( x ) as in §18.3, satisfies the Toda equation …
Random Matrix Theory
Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …
6: 18.5 Explicit Representations
Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).
p n ( x ) F ( x ) κ n
H n ( x ) 1 ( - 1 ) n
H 0 ( x ) = 1 ,
H 1 ( x ) = 2 x ,
He 0 ( x ) = 1 ,
He 1 ( x ) = x ,
7: 18.8 Differential Equations
Table 18.8.1: Classical OP’s: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
f ( x ) A ( x ) B ( x ) C ( x ) λ n
H n ( x ) 1 - 2 x 0 2 n
e - 1 2 x 2 H n ( x ) 1 0 - x 2 2 n + 1
He n ( x ) 1 - x 0 n
8: 18.17 Integrals
Hermite
Hermite
Hermite
Hermite
Hermite
9: 18.7 Interrelations and Limit Relations
§18.7 Interrelations and Limit Relations
18.7.11 He n ( x ) = 2 - 1 2 n H n ( 2 - 1 2 x ) ,
18.7.12 H n ( x ) = 2 1 2 n He n ( 2 1 2 x ) .
18.7.19 H 2 n ( x ) = ( - 1 ) n 2 2 n n ! L n ( - 1 2 ) ( x 2 ) ,
10: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. …
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 )
H n ( x ) ( - 1 ) n H n ( x ) ( - 1 ) n ( n + 1 ) n 2 ( - 1 ) n ( n + 1 ) n + 1
He n ( x ) ( - 1 ) n He n ( x ) ( - 1 2 ) n ( n + 1 ) n ( - 1 2 ) n ( n + 1 ) n + 1