continuous%20q-ultraspherical%20polynomials
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11: 18.26 Wilson Class: Continued
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Wilson Continuous Dual Hahn
… ►Wilson Continuous Hahn
… ►Continuous Dual Hahn Meixner–Pollaczek
… ►Continuous Dual Hahn
… ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …12: 18.28 Askey–Wilson Class
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§18.28(v) Continuous -Ultraspherical Polynomials
… ►§18.28(vi) Continuous -Hermite Polynomials
… ►§18.28(vii) Continuous -Hermite Polynomials
… ►For continuous -Hermite polynomials the orthogonality measure is not unique. … ►§18.28(ix) Continuous -Jacobi Polynomials
…13: 32.16 Physical Applications
14: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Continuous Hahn polynomials.
J. Phys. A 18 (16), pp. L1017–L1019.
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Continuous
-Hermite Polynomials when
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In
-series and Partitions (Minneapolis, MN, 1988),
IMA Vol. Math. Appl., Vol. 18, pp. 151–158.
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15: 18.19 Hahn Class: Definitions
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►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek).
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Hahn class (or linear lattice class). These are OP’s where the role of is played by or or (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.
Wilson class (or quadratic lattice class). These are OP’s ( of degree in , quadratic in ) where the role of the differentiation operator is played by or or . The Wilson class consists of two discrete and two continuous families.
Continuous Hahn
… ► …16: Wolter Groenevelt
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►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems.
►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials.
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17: 36.5 Stokes Sets
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36.5.4
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36.5.7
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►For the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation
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►In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets).
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18: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►When is absolutely continuous, i.
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§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases
… ►and completeness relation … ► … ►§1.18(vii) Continuous Spectra: More General Cases
…19: 18.21 Hahn Class: Interrelations
§18.21 Hahn Class: Interrelations
►§18.21(i) Dualities
… ►§18.21(ii) Limit Relations and Special Cases
… ►Hahn Jacobi
… ►Continuous Hahn Meixner–Pollaczek
…20: 8 Incomplete Gamma and Related
Functions
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