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Boole summation formula

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1: 24.17 Mathematical Applications
§24.17(i) Summation
Euler–Maclaurin Summation Formula
Boole Summation Formula
2: Bibliography H
  • A. J. S. Hamilton (2001) Formulae for growth factors in expanding universes containing matter and a cosmological constant. Monthly Notices Roy. Astronom. Soc. 322 (2), pp. 419–425.
  • G. H. Hardy and S. Ramanujan (1918) Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, pp. 75–115.
  • M. Hauss (1998) A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.
  • K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
  • F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.
  • 3: 1.8 Fourier Series
    Parseval’s Formula
    Poisson’s Summation Formula
    1.8.16 n = - e - ( n + x ) 2 ω = π ω ( 1 + 2 n = 1 e - n 2 π 2 / ω cos ( 2 n π x ) ) , ω > 0 .
    4: 5.19 Mathematical Applications
    §5.19(i) Summation of Rational Functions
    5: 28.34 Methods of Computation
  • (b)

    Representations for w I ( π ; a , ± q ) with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed a and q ; see Schäfke (1961a).

  • (a)

    Summation of the power series in §§28.6(i) and 28.15(i) when | q | is small.

  • (a)

    Summation of the power series in §§28.6(ii) and 28.15(ii) when | q | is small.

  • (a)

    Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of q and z .

  • 6: 17.8 Special Cases of ψ r r Functions
    Ramanujan’s ψ 1 1 Summation
    Bailey’s Bilateral Summations
    For similar formulas see Verma and Jain (1983).
    7: 2.10 Sums and Sequences
    §2.10(i) Euler–Maclaurin Formula
    This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula: …
    §2.10(ii) Summation by Parts
    The formula for summation by parts is …
    8: Bibliography Z
  • Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
  • I. J. Zucker (1979) The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.
  • 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.
  • 9: 3.5 Quadrature
    See also Poisson’s summation formula1.8(iv)). …
    Gauss–Legendre Formula
    Gauss–Laguerre Formula
    a complex Gauss quadrature formula is available. …
    10: Bibliography W
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • E. J. Weniger (1989) Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.
  • J. Wimp (1968) Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.
  • J. Wimp (1987) Explicit formulas for the associated Jacobi polynomials and some applications. Canad. J. Math. 39 (4), pp. 983–1000.
  • R. Wong (1982) Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.