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Gauss–Hermite formula


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1: 3.5 Quadrature
GaussHermite Formula
Table 3.5.10: Nodes and weights for the 5-point GaussHermite formula.
± x k w k
Table 3.5.11: Nodes and weights for the 10-point GaussHermite formula.
± x k w k
Table 3.5.12: Nodes and weights for the 15-point GaussHermite formula.
± x k w k
Table 3.5.13: Nodes and weights for the 20-point GaussHermite formula.
± x k w k
2: 18.5 Explicit Representations
§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). …
3: Bibliography G
  • C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).
  • W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
  • W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 4: Bibliography R
  • I. S. Reed, D. W. Tufts, X. Yu, T. K. Truong, M. T. Shih, and X. Yin (1990) Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
  • H. Rosengren (1999) Another proof of the triple sum formula for Wigner 9 j -symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
  • R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.
  • 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 I
  • M. E. H. Ismail and D. R. Masson (1994) q -Hermite polynomials, biorthogonal rational functions, and q -beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
  • A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
  • A. R. Its and A. A. Kapaev (1998) Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.
  • K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida (1991) From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.