Hurwitz criterion for stable polynomials
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21—30 of 277 matching pages
21: Bibliography H
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Lamé polynomials of large order.
SIAM J. Math. Anal. 8 (5), pp. 800–842.
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Orthogonal Laurent polynomials.
Nederl. Akad. Wetensch. Indag. Math. 48 (1), pp. 17–36.
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Roots of the Euler polynomials.
Pacific J. Math. 64 (1), pp. 181–191.
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Bernoulli numbers and polynomials via residues.
J. Number Theory 76 (2), pp. 178–193.
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Einige Eigenschaften der Dirichletschen Functionen , die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten.
Zeitschrift für Math. u. Physik 27, pp. 86–101 (German).
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22: Bibliography C
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A note on Euler numbers and polynomials.
Nagoya Math. J. 7, pp. 35–43.
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Stability properties of disk polynomials.
Numer. Algorithms.
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Properties of generalized Freud polynomials.
J. Approx. Theory 225, pp. 148–175.
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On some series representations of the Hurwitz zeta function.
J. Comput. Appl. Math. 216 (1), pp. 297–305.
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An efficient algorithm for the Hurwitz zeta and related functions.
J. Comput. Appl. Math. 225 (2), pp. 338–346.
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23: Bibliography K
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The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers.
Math. Ann. 216 (1), pp. 1–4.
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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials.
Appl. Anal. 90 (3-4), pp. 731–746.
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The addition formula for Laguerre polynomials.
SIAM J. Math. Anal. 8 (3), pp. 535–540.
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Meixner-Pollaczek polynomials and the Heisenberg algebra.
J. Math. Phys. 30 (4), pp. 767–769.
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Askey-Wilson polynomial.
Scholarpedia 7 (7), pp. 7761.
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24: Software Index
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18 Orthogonal Polynomials | |||||||||||||||||||||||||
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24 Bernoulli and Euler Polynomials | |||||||||||||||||||||||||
24.21(ii) , , , | ✓ | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Derive, MuPAD | ||||||||||||
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25.21(iv) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | a | |||||||||||||||
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25: 25.15 Dirichlet -functions
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25.15.3
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26: 25.12 Polylogarithms
27: Bibliography M
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Symmetric Functions and Orthogonal Polynomials.
University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.
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Orthogonal polynomials associated with root systems.
Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).
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Affine Hecke Algebras and Orthogonal Polynomials.
Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.
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Derivatives of the Hurwitz zeta function for rational arguments.
J. Comput. Appl. Math. 100 (2), pp. 201–206.
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Exceptional orthogonal polynomials.
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28: 14.32 Methods of Computation
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►In particular, for small or moderate values of the parameters and the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real.
In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967).
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29: 18.40 Methods of Computation
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§18.40(i) Computation of Polynomials
►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). … … ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let be a positive integer and define …See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. …30: 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.
…In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’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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Equations (25.11.6), (25.11.19), and (25.11.20)
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Originally all six integrands in these equations were incorrect because their numerators contained the function . The correct function is . The new equations are:
25.11.6
, ,
Reported 2016-05-08 by Clemens Heuberger.
25.11.19
, ,
Reported 2016-06-27 by Gergő Nemes.
25.11.20
, ,
Reported 2016-06-27 by Gergő Nemes.