of a polynomial
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11: 13.6 Relations to Other Functions
12: 35.4 Partitions and Zonal Polynomials
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35.4.2
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►Therefore is a symmetric polynomial in the eigenvalues of .
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35.4.6
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35.4.8
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35.4.9
13: 18.19 Hahn Class: Definitions
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►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
.
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Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.
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►These polynomials are orthogonal on , and with , are defined as follows.
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►A special case of (18.19.8) is .
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Charlier | , |
14: Roelof Koekoek
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►Koekoek is mainly a teacher of mathematics and has published a few papers on orthogonal polynomials.
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15: 29.17 Other Solutions
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►If (29.2.1) admits a Lamé polynomial solution , then a second linearly independent solution is given by
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16: 33.14 Definitions and Basic Properties
17: 8.20 Asymptotic Expansions of
18: 31.8 Solutions via Quadratures
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31.8.2
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►Here is a polynomial of degree in and of degree in , that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation.
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19: 31.15 Stieltjes Polynomials
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►where is a polynomial of degree not exceeding .
There exist at most
polynomials
of degree not exceeding such that for , (31.15.1) has a polynomial solution of degree .
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►If are the zeros of an th degree Stieltjes polynomial
, then every zero is either one of the parameters or a solution of the system of equations
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►If is a zero of the Van Vleck polynomial
, corresponding to an th degree Stieltjes polynomial
, and are the zeros of (the derivative of ), then is either a zero of or a solution of the equation
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►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index , where each is a nonnegative integer, there is a unique Stieltjes polynomial with zeros in the open interval for each .
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