kernel polynomials
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1: 18.2 General Orthogonal Polynomials
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18.2.12
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18.2.13
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Kernel Polynomials
… βΊThen the kernel polynomials … βΊPoisson kernel
…2: 18.18 Sums
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§18.18(vii) Poisson Kernels
… βΊLaguerre
… βΊHermite
… βΊFor the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938). …3: 31.10 Integral Equations and Representations
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Kernel Functions
… βΊThe kernel must satisfy … βΊKernel Functions
… βΊThe kernel must satisfy … βΊleads to the kernel equation …4: Bibliography
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SzegΕ Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials.
Comm. Math. Phys. 398 (3), pp. 1291–1348.
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5: 10.69 Uniform Asymptotic Expansions for Large Order
6: 10.67 Asymptotic Expansions for Large Argument
7: 18.12 Generating Functions
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Jacobi
… βΊUltraspherical
… βΊLegendre
… βΊLaguerre
… βΊSee §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.8: Errata
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βΊWe have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials.
…In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and SzegΕ’s theorem.
We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials.
We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials.
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Equation (18.35.1)
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18.35.1
These equations which were previously given for Pollaczek polynomials of type 2 has been updated for Pollaczek polynomials of type 3.
9: Bibliography B
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The generating function of Jacobi polynomials.
J. London Math. Soc. 13, pp. 8–12.
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A generalisation of the Legendre polynomial.
Proc. London Math. Soc. (2) 3 (3), pp. 111–123.
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Polynomials defined by a difference system.
J. Math. Anal. Appl. 2 (2), pp. 223–263.
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Tables of Normalized Associated Legendre Polynomials.
Pergamon Press, The Macmillan Co., Oxford-New York.
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Integral transforms with generalized Legendre functions as kernels.
Compositio Math. 18, pp. 235–287.
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10: Bibliography J
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Fonctions de Mathieu et polynômes de Klein-Gordon.
C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).
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A note on sampling expansion for a transform with parabolic cylinder kernel.
Inform. Sci. 26 (2), pp. 155–158.
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Uniform asymptotic expansions for Meixner polynomials.
Constr. Approx. 14 (1), pp. 113–150.
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Asymptotic formulas for the zeros of the Meixner polynomials.
J. Approx. Theory 96 (2), pp. 281–300.
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