Rogers–Szegő polynomials
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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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31.5.2
►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities .
These solutions are the Heun polynomials.
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2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
… ►Normalization
… ►Orthogonal Invariance
… ►Summation
… ►Mean-Value
…3: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …4: 18.3 Definitions
§18.3 Definitions
… ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to for these polynomials and for coefficients up to for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►Bessel polynomials
►Bessel polynomials are often included among the classical OP’s. …5: 18.1 Notation
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Classical OP’s
… ►Hahn Class OP’s
… ►Wilson Class OP’s
… ►Nor do we consider the shifted Jacobi polynomials: …or the dilated Chebyshev polynomials of the first and second kinds: …6: 17.18 Methods of Computation
7: David M. Bressoud
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►His books are Analytic and Combinatorial
Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No.
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8: 18.33 Polynomials Orthogonal on the Unit Circle
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Askey
… ►When the Askey case is also known as the Rogers–Szegő case. See for a more general class Costa et al. (2012). … ►Then the corresponding orthonormal polynomials are … ►For a polynomial …9: 26.10 Integer Partitions: Other Restrictions
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26.10.3
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§26.10(iv) Identities
►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …10: Bibliography
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Some orthogonal -polynomials.
Math. Nachr. 30, pp. 47–61.
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Zeros of Stieltjes and Van Vleck polynomials.
Trans. Amer. Math. Soc. 252, pp. 197–204.
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-identities of Auluck, Carlitz, and Rogers.
Duke Math. J. 33 (3), pp. 575–581.
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Multiple series Rogers-Ramanujan type identities.
Pacific J. Math. 114 (2), pp. 267–283.
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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
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