Hermite
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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)). … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of for Jacobi polynomials, in powers of for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for are given in §18.5(iv). …2: 18.41 Tables
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►For () see §14.33.
►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates , , , and for .
The ranges of are for and , and for and .
The precision is 10D, except for which is 6-11S.
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►For , , and see §3.5(v).
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3: 7.10 Derivatives
4: 18.36 Miscellaneous Polynomials
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►They are related to Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers.
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Type III -Hermite EOP’s
►Hermite EOP’s are defined in terms of classical Hermite OP’s. The type III -Hermite EOP’s, missing polynomial orders and , are the complete set of polynomials, with real coefficients and defined explicitly as … ►In §18.39(i) it is seen that the functions, , 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: 12.1 Special Notation
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►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values.
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►Whittaker’s notation is useful when is a nonnegative integer (Hermite polynomial case).
6: 28.9 Zeros
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►For the zeros of and approach asymptotically the zeros of , and the zeros of and approach asymptotically the zeros of .
Here denotes the Hermite polynomial of degree (§18.3).
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7: 18.4 Graphics
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8: 18.6 Symmetry, Special Values, and Limits to Monomials
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►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1.
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