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Gauss–Chebyshev formula

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1: 3.5 Quadrature
GaussChebyshev Formula
2: 18.5 Explicit Representations
Chebyshev
§18.5(ii) Rodrigues Formulas
Related formula: … For corresponding formulas for Chebyshev, Legendre, and the Hermite He n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
Chebyshev
3: 18.12 Generating Functions
and a similar formula by symmetry; compare the second row in Table 18.6.1. For the hypergeometric function F 1 2 see §§15.1, 15.2(i). …
Chebyshev
18.12.7 1 - z 2 1 - 2 x z + z 2 = 1 + 2 n = 1 T n ( x ) z n , | z | < 1 .
18.12.8 1 - x z 1 - 2 x z + z 2 = n = 0 T n ( x ) z n , | z | < 1 .
4: Bibliography R
  • A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.
  • M. Razaz and J. L. Schonfelder (1980) High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.
  • M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.
  • E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
  • 5: Bibliography T
  • Y. Takei (1995) On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.
  • N. M. Temme (2003) Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
  • P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
  • L. N. Trefethen (2008) Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.
  • A. Trellakis, A. T. Galick, and U. Ravaioli (1997) Rational Chebyshev approximation for the Fermi-Dirac integral F - 3 / 2 ( x ) . Solid–State Electronics 41 (5), pp. 771–773.
  • 6: Bibliography Z
  • A. Zarzo, J. S. Dehesa, and R. J. Yañez (1995) Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.
  • J. Zhang and J. A. Belward (1997) Chebyshev series approximations for the Bessel function Y n ( z ) of complex argument. Appl. Math. Comput. 88 (2-3), pp. 275–286.
  • M. I. Žurina and L. N. Karmazina (1966) Tables and formulae for the spherical functions P - 1 / 2 + i τ m ( z ) . Translated by E. L. Albasiny, Pergamon Press, Oxford.
  • 7: Bibliography M
  • A. J. MacLeod (1993) Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.
  • J. C. Mason and D. C. Handscomb (2003) Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton, FL.
  • J. C. Mason (1993) Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. In Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), Vol. 49, pp. 169–178.
  • G. F. Miller (1966) On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.