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1: 18.3 Definitions
§18.3 Definitions
The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). …
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, 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
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: 18.36 Miscellaneous Polynomials
The type III X 2 -Hermite EOP’s, missing polynomial orders 1 and 2 , are the complete set of polynomials, with real coefficients and defined explicitly as
18.36.8 H ^ 0 ( x ) = 2 3 / 2 π 1 / 4 ,
18.36.9 H ^ n + 3 ( x ) = ( 4 x 2 + 2 ) H n + 1 ( x ) + 8 x H n ( x ) π 1 / 4 2 n + 1 ( n + 3 ) n ! = 𝒲 { H 1 ( x ) , H 2 ( x ) , H n + 3 ( x ) } π 1 / 4 2 n + 7 ( n + 1 ) ( n + 2 ) ( n + 3 ) ! , n = 0 , 1 , ,
In §18.39(i) it is seen that the functions, w ( x ) H ^ n + 3 ( x ) , are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …
5: 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). …
6: 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
12 H n ( x ) 1 2 x 0 2 n
13 e 1 2 x 2 H n ( x ) 1 0 x 2 2 n + 1
14 𝐻𝑒 n ( x ) 1 x 0 n
7: 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 ) ,
18.7.20 H 2 n + 1 ( x ) = ( 1 ) n 2 2 n + 1 n ! x L n ( 1 2 ) ( x 2 ) .
8: 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
9: 18.17 Integrals
Hermite
Hermite
Hermite
Hermite
Hermite
10: 12.7 Relations to Other Functions
§12.7(i) Hermite Polynomials
12.7.2 U ( n 1 2 , z ) = D n ( z ) = e 1 4 z 2 𝐻𝑒 n ( z ) = 2 n / 2 e 1 4 z 2 H n ( z / 2 ) , n = 0 , 1 , 2 , ,
12.7.3 V ( n + 1 2 , z ) = 2 / π e 1 4 z 2 ( i ) n 𝐻𝑒 n ( i z ) = 2 / π e 1 4 z 2 ( i ) n 2 1 2 n H n ( i z / 2 ) , n = 0 , 1 , 2 , .