continuous q-ultraspherical polynomials
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11: 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
…12: 18.22 Hahn Class: Recurrence Relations and Differences
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§18.22(i) Recurrence Relations in
… ►Continuous Hahn
… ►Continuous Hahn
… ►§18.22(iii) -Differences
… ►Continuous Hahn
…13: 18.20 Hahn Class: Explicit Representations
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§18.20(i) Rodrigues Formulas
… ►For the Hahn polynomials and … ►Continuous Hahn
… ►§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions
… ►(For symmetry properties of with respect to , , , see Andrews et al. (1999, Corollary 3.3.4).) …14: 32.16 Physical Applications
15: 18.23 Hahn Class: Generating Functions
16: 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
…17: 1.17 Integral and Series Representations of the Dirac Delta
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►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that
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►for all functions that are continuous when , and for each , converges absolutely for all sufficiently large values of .
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►More generally, assume is piecewise continuous (§1.4(ii)) when for any finite positive real value of , and for each , converges absolutely for all sufficiently large values of .
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►provided that is continuous when , and for each , converges absolutely for all sufficiently large values of (as in the case of (1.17.6)).
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►provided that is continuous and of period ; see §1.8(ii).
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18: 28.9 Zeros
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►They are continuous in .
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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19: 18.18 Sums
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►Alternatively, assume is real and continuous and is piecewise continuous on .
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►Assume is real and continuous and is piecewise continuous on .
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►Assume is real and continuous and is piecewise continuous on .
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Ultraspherical
… ►Legendre
…20: 18.37 Classical OP’s in Two or More Variables
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