via classical orthogonal polynomials
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1: 18.38 Mathematical Applications
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Quadrature
… ►Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
…2: 18.3 Definitions
§18.3 Definitions
… ►As given by a Rodrigues formula (18.5.5).
3: 18.36 Miscellaneous Polynomials
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§18.36(ii) Sobolev Orthogonal Polynomials
… ►§18.36(iv) Orthogonal Matrix Polynomials
… ►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, for as per (18.2.9_5). … ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the polynomials, self-adjointness implying both orthogonality and completeness. … ►§18.36(vi) Exceptional Orthogonal Polynomials
…4: 18.1 Notation
5: 18.2 General Orthogonal Polynomials
§18.2 General Orthogonal Polynomials
… ►The classical orthogonal polynomials are defined with: … ► … ►§18.2(vi) Zeros
… ► …6: Bibliography I
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On polynomials orthogonal with respect to certain Sobolev inner products.
J. Approx. Theory 65 (2), pp. 151–175.
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Two families of orthogonal polynomials related to Jacobi polynomials.
Rocky Mountain J. Math. 21 (1), pp. 359–375.
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An electrostatics model for zeros of general orthogonal polynomials.
Pacific J. Math. 193 (2), pp. 355–369.
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Classical and Quantum Orthogonal Polynomials in One Variable.
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
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Classical and Quantum Orthogonal Polynomials in One Variable.
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
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7: 18.15 Asymptotic Approximations
§18.15 Asymptotic Approximations
►§18.15(i) Jacobi
… ►With the expansions in Chapter 12 are for the parabolic cylinder function , which is related to the Hermite polynomials via … ►The asymptotic behavior of the classical OP’s as with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. … ►See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).8: 18.19 Hahn Class: Definitions
§18.19 Hahn Class: Definitions
… ►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 . … ►These polynomials are orthogonal on , and are defined as follows. …A special case of (18.19.8) is .9: 18.18 Sums
§18.18 Sums
… ►§18.18(ii) Addition Theorems
… ►§18.18(iii) Multiplication Theorems
… ►§18.18(v) Linearization Formulas
… ►10: 18.40 Methods of Computation
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►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)).
…For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev.
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