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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 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.
  • 5: 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.
  • 6: 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.
  • 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.
  • S. C. Milne (1988) A q -analog of the Gauss summation theorem for hypergeometric series in U ( n ) . Adv. in Math. 72 (1), pp. 59–131.
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.