Hermite polynomial case
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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
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Jacobi
… ►Ultraspherical
… ►Legendre
… ►Hermite
… ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.13: 18.18 Sums
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 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 , Krawtchouk polynomials , Meixner polynomials , and Charlier polynomials . … ►A special case of (18.19.8) is .15: 8.11 Asymptotic Approximations and Expansions
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►This reference also contains explicit formulas for in terms of Stirling numbers and for the case
an asymptotic expansion for as .
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►In the case that , a positive integer, the -region of validity of (8.11.7) is discussed in Ameur and Cronvall (2023).
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►in both cases uniformly with respect to bounded real values of .
…For related expansions involving Hermite polynomials see Pagurova (1965).
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16: 18.8 Differential Equations
17: 18.39 Applications in the Physical Sciences
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►Here the are Hermite polynomials, , and .
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►c) A Rational SUSY Potential
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►and eigenvalues , with as above, with the weight function of (18.36.10), and a type III Hermite EOP defined by (18.36.8) and (18.36.9).
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►This seems odd at first glance as is a polynomial of order for , seemingly suggesting that for , this being the first excited state, i.
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►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3).
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18: 18.30 Associated OP’s
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►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 -axis each multiplied by the polynomial product evaluated at the corresponding values of , as in (18.2.3).
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§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 of order with respect to the 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
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►The eigenfunction expansions of (1.8.1) and (1.8.2) follow from Cases 1, 2, above.
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