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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 Orthogonal Polynomials
Chapter 18 Orthogonal Polynomials
…6: 18.28 Askey–Wilson Class
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§18.28(iii) Al-Salam–Chihara Polynomials
… ►§18.28(iv) -Al-Salam–Chihara Polynomials
… ►For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006). ►§18.28(v) Continuous -Ultraspherical Polynomials
… ►These polynomials are also called Rogers polynomials. …7: 18.1 Notation
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►
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►Nor do we consider the shifted Jacobi polynomials:
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Classical OP’s
… ►Hahn Class OP’s
… ►Wilson Class OP’s
… ►Al-Salam–Chihara: .
8: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Orthogonal polynomials.
Mem. Amer. Math. Soc. 18 (213), pp. v+185 pp..
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Géza Freud, orthogonal polynomials and Christoffel functions. A case study.
J. Approx. Theory 48 (1), pp. 3–167.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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9: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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On the zeros of the Askey-Wilson polynomials, with applications to coding theory.
SIAM J. Math. Anal. 18 (1), pp. 191–207.
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An Introduction to Orthogonal Polynomials.
Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York.
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Extremal measures for a system of orthogonal polynomials.
Constr. Approx. 9, pp. 111–119.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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10: 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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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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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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