Hurwitz criterion for stable polynomials
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1: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
►§25.11(i) Definition
… ►§25.11(ii) Graphics
… ►§25.11(vi) Derivatives
… ►§25.11(ix) Integrals
…2: 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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3: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
… ►Normalization
… ►Orthogonal Invariance
… ►Summation
… ►Mean-Value
…4: 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). …5: 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. …6: 1.11 Zeros of Polynomials
§1.11 Zeros of Polynomials
… ►§1.11(ii) Elementary Properties
… ►§1.11(v) Stable Polynomials
… ►with real coefficients, is called stable if the real parts of all the zeros are strictly negative. ►Hurwitz Criterion
…7: 25.1 Special Notation
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►The main related functions are the Hurwitz zeta function , the dilogarithm , the polylogarithm (also known as Jonquière’s function ), Lerch’s transcendent , and the Dirichlet -functions .
nonnegative integers. | |
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Bernoulli number and polynomial (§24.2(i)). | |
periodic Bernoulli function . | |
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8: 21.10 Methods of Computation
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Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.