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representations by Euler–Maclaurin formula

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1: 25.2 Definition and Expansions
§25.2(iii) Representations by the EulerMaclaurin Formula
2: 25.11 Hurwitz Zeta Function
§25.11(iii) Representations by the EulerMaclaurin Formula
3: Bibliography E
  • G. P. Egorychev (1984) Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.
  • D. Elliott (1998) The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).
  • T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.
  • L. Euler (1768) Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.
  • J. A. Ewell (1990) A new series representation for ζ ( 3 ) . Amer. Math. Monthly 97 (3), pp. 219–220.
  • 4: Bibliography B
  • Á. Baricz and T. K. Pogány (2013) Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.
  • W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
  • P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.
  • 5: 9.12 Scorer Functions
    where …
    §9.12(v) Connection Formulas
    §9.12(vi) Maclaurin Series
    §9.12(vii) Integral Representations
    6: Bibliography F
  • P. Flajolet and B. Salvy (1998) Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.
  • J. P. M. Flude (1998) The Edmonds asymptotic formulas for the 3 j and 6 j symbols. J. Math. Phys. 39 (7), pp. 3906–3915.
  • W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.
  • G. Freud (1976) On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.
  • 7: 13.29 Methods of Computation
    Although the Maclaurin series expansion (13.2.2) converges for all finite values of z , it is cumbersome to use when | z | is large owing to slowness of convergence and cancellation. … For U ( a , b , z ) and W κ , μ ( z ) we may integrate along outward rays from the origin in the sectors 1 2 π < | ph z | < 3 2 π , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). …
    §13.29(iii) Integral Representations
    The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. …
    13.29.8 w ( n ) π e 1 2 z z 1 4 ( 4 a - 2 b + 1 ) Γ ( a ) Γ ( a + 1 - b ) n 1 4 ( 4 a - 2 b - 3 ) e - 2 n z ,
    8: 18.17 Integrals
    §18.17(ii) Integral Representations for Products
    Ultraspherical
    Legendre
    For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). … and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. …