denominator polynomials
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8 matching pages
1: 18.30 Associated OP’s
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Numerator and Denominator Polynomials
►The are also referred to as the numerator polynomials, the then being the denominator polynomials, in that the -th approximant of the continued fraction, , … ►Markov’s Theorem
►The ratio , as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. …2: 18.2 General Orthogonal Polynomials
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►Because of (18.2.36) the OP’s are also called monic denominator
polynomials and the OP’s , or, equivalently, the , are called the monic numerator polynomials.
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3: 18.13 Continued Fractions
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is the denominator of the th approximant to:
…and is the denominator of the th approximant to:
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is the denominator of the th approximant to:
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is the denominator of the th approximant to:
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is the denominator of the th approximant to:
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4: Errata
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►We 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.
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5: 18.39 Applications in the Physical Sciences
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►The associated Coulomb–Laguerre polynomials are defined as
…see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an in the denominator.
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§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods
… ►The Coulomb–Pollaczek Polynomials
… ►§18.39(v) Other Applications
…6: 1.2 Elementary Algebra
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►To find the polynomials
, , multiply both sides by the denominator of the left-hand side and equate coefficients.
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7: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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►The generalized hypergeometric function with matrix argument , numerator parameters , and denominator parameters is
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