of a polynomial
(0.017 seconds)
1—10 of 232 matching pages
1: 18.31 Bernstein–Szegő Polynomials
2: 15.13 Zeros
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►If , , , , or , then is not defined, or reduces to a polynomial, or reduces to times a polynomial.
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3: 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 .
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4: 10.19 Asymptotic Expansions for Large Order
5: 18.34 Bessel Polynomials
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►With the notation of Koekoek et al. (2010, (9.13.1)) the left-hand side of (18.34.1) has to be replaced by .
…where is a modified spherical Bessel function (10.49.9), and
… Sometimes the polynomials
are called reverse Bessel polynomials.
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►Hence the full system of polynomials
cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if :
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►For uniform asymptotic expansions of as in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c).
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6: 17.17 Physical Applications
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►In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role.
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►They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics.
See Kassel (1995).
►A substantial literature on -deformed quantum-mechanical Schrödinger equations has developed recently.
It involves -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials.
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7: 8.7 Series Expansions
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►
8.7.6
, .
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8: 18.21 Hahn Class: Interrelations
9: 1.11 Zeros of Polynomials
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►A polynomial of degree with real or complex coefficients has exactly real or complex zeros counting multiplicity.
…A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative.
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►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity.
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►The discriminant of is defined by
…The elementary
symmetric functions of the zeros are (with )
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10: 18.35 Pollaczek Polynomials
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►The type 2 polynomials reduce for to ultraspherical polynomials, see (18.35.8).
►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8))
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►More generally, the are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)).
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►See Bo and Wong (1996) for an asymptotic expansion of as , with and fixed.
…Also included is an asymptotic approximation for the zeros of .
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