in inverse factorial series
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1: 6.18 Methods of Computation
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►For large and , expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available.
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2: 10.41 Asymptotic Expansions for Large Order
3: 2.9 Difference Equations
4: 6.10 Other Series Expansions
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§6.10(i) Inverse Factorial Series
…5: 2.8 Differential Equations with a Parameter
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►In Case I there are no transition points in
and is analytic.
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►The same approach is used in all three cases.
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►These are elementary functions in Case I, and Airy functions (§9.2) in Case II.
In Case III the approximating equation is
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►For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004).
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6: 4.24 Inverse Trigonometric Functions: Further Properties
§4.24 Inverse Trigonometric Functions: Further Properties
►§4.24(i) Power Series
… ►§4.24(ii) Derivatives
… ►§4.24(iii) Addition Formulas
… ►The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. …7: Bibliography O
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Summing one- and two-dimensional series related to the Euler series.
J. Comput. Appl. Math. 98 (2), pp. 245–271.
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Inverse factorial-series solutions of difference equations.
Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations.
Methods Appl. Anal. 1 (1), pp. 1–13.
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The Generalized Exponential Integral.
In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.),
International Series of Numerical Mathematics, Vol. 119, pp. 497–510.
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8: 24.19 Methods of Computation
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►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
We list here three methods, arranged in increasing order of efficiency.
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Tanner and Wagstaff (1987) derives a congruence for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).