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expansion in inverse factorials

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1: 6.10 Other Series Expansions
§6.10(i) Inverse Factorial Series
2: 6.18 Methods of Computation
For large x and | z | , expansions in inverse factorial series (§6.10(i)) or asymptotic expansions6.12) are available. …
3: 10.41 Asymptotic Expansions for Large Order
For expansions in inverse factorial series see Dunster et al. (1993). …
4: 2.9 Difference Equations
For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). …
5: 24.19 Methods of Computation
  • Buhler et al. (1992) uses the expansion

    24.19.3 t 2 cosh t 1 = 2 n = 0 ( 2 n 1 ) B 2 n t 2 n ( 2 n ) ! ,

    and computes inverses modulo p of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

  • 6: 2.8 Differential Equations with a Parameter
    The form of the asymptotic expansion depends on the nature of the transition points in 𝐃 , that is, points at which f ( z ) has a zero or singularity. … For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). … For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. … For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … However, in all cases with λ > 2 and λ 0 or ± 1 , only uniform asymptotic approximations are available, not uniform asymptotic expansions. …
    7: Bibliography O
  • F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.
  • A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.
  • A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.
  • F. W. J. Olver (1964b) Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
  • F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.
  • 8: 18.18 Sums
    Legendre
    Laguerre
    Hermite
    Ultraspherical
    Hermite
    9: 22.10 Maclaurin Series
    §22.10(i) Maclaurin Series in z
    Further terms may be derived by substituting in the differential equations (22.13.13), (22.13.14), (22.13.15). The full expansions converge when | z | < min ( K ( k ) , K ( k ) ) .
    §22.10(ii) Maclaurin Series in k and k
    Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …
    10: 18.39 Applications in the Physical Sciences
    §18.39 Applications in the Physical Sciences
    see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an n ! in the denominator. …
    §18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
    Discretized and Continuum Expansions of Scattering Eigenfunctions in terms of Pollaczek Polynomials: J-matrix Theory
    The equivalent quadrature weight, w i / w CP ( x i ) , also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii). …