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11: 18.21 Hahn Class: Interrelations
§18.21 Hahn Class: Interrelations
§18.21(i) Dualities
§18.21(ii) Limit Relations and Special Cases
Charlier Hermite
Meixner–Pollaczek Hermite
12: 18.12 Generating Functions
Jacobi
Ultraspherical
Legendre
Hermite
See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.
13: 18.18 Sums
Hermite
Hermite
Hermite
Hermite
Hermite
14: 18.19 Hahn Class: Definitions
§18.19 Hahn Class: Definitions
The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator d d x in the case of the classical OP’s is played by a suitable difference operator. …
Hahn, Krawtchouk, Meixner, and Charlier
Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials Q n ( x ; α , β , N ) , Krawtchouk polynomials K n ( x ; p , N ) , Meixner polynomials M n ( x ; β , c ) , and Charlier polynomials C n ( x ; a ) . … A special case of (18.19.8) is w ( 1 / 2 ) ( x ; π / 2 ) = π cosh ( π x ) .
15: 8.11 Asymptotic Approximations and Expansions
This reference also contains explicit formulas for b k ( λ ) in terms of Stirling numbers and for the case λ > 1 an asymptotic expansion for b k ( λ ) as k . … In the case that a = n , a positive integer, the z -region of validity of (8.11.7) is discussed in Ameur and Cronvall (2023). … in both cases uniformly with respect to bounded real values of y . …For related expansions involving Hermite polynomials see Pagurova (1965). …
16: 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
8 L n ( α ) ( x ) x α + 1 x 0 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
17: 18.39 Applications in the Physical Sciences
Here the H n ( x ) are Hermite polynomials, w ( x ) = e x 2 , and h n = 2 n n ! π . … c) A Rational SUSY Potential argumentand eigenvalues n + 3 , with n as above, with w ( x ) the weight function of (18.36.10), and H ^ n + 3 ( x ) a type III Hermite EOP defined by (18.36.8) and (18.36.9). … This seems odd at first glance as H ^ n + 3 ( x ) is a polynomial of order n + 3 for n = 0 , 1 , 2 , , seemingly suggesting that for n = 0 , this being the first excited state, i. … These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). …
18: 18.30 Associated OP’s
For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real x -axis each multiplied by the polynomial product evaluated at the corresponding values of x , as in (18.2.3).
§18.30(iv) Associated Hermite Polynomials
The recursion relation for the associated Hermite polynomials, see (18.30.2), and (18.30.3), is …
Numerator and Denominator Polynomials
In the monic case, the monic associated polynomials p ^ n ( x ; c ) of order c with respect to the p ^ n ( x ) are obtained by simply changing the initialization and recursions, respectively, of (18.30.2) and (18.30.3) to …
19: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
The eigenfunction expansions of (1.8.1) and (1.8.2) follow from Cases 1, 2, above. …
Hermite’s Differential Equation, X = ( , )
Writing Hermite’s differential equation (see Tables 18.3.1 and 18.8.1) in the form above, the eigenfunctions are e x 2 / 2 H n ( x ) ( H n a Hermite polynomial, n = 0 , 1 , 2 , ), with eigenvalues λ n = 2 n + 1 𝝈 p , for the differential operator …
§1.18(vii) Continuous Spectra: More General Cases
By Weyl’s alternative n 1 equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for n 2 . …
20: 18.5 Explicit Representations
§18.5 Explicit Representations
Hermite
For corresponding formulas for Chebyshev, Legendre, and the Hermite 𝐻𝑒 n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
Hermite