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1: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 3.6 Linear Difference Equations
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►with , , can be computed recursively for .
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Example 2. Weber Function
… ►for , and as … ►In the notation of §3.6(v) we have and . … ►For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).3: Bibliography C
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Some congruences for the Bernoulli numbers.
Amer. J. Math. 75 (1), pp. 163–172.
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-Bernoulli and Eulerian numbers.
Trans. Amer. Math. Soc. 76 (2), pp. 332–350.
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Expansions of -Bernoulli numbers.
Duke Math. J. 25 (2), pp. 355–364.
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A Unified Approach to Recurrence Algorithms.
In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.),
International Series of Computational Mathematics, Vol. 119, pp. 97–120.
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4: DLMF Project News
error generating summary5: 10.74 Methods of Computation
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►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation.
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►And since there are no error terms they could, in theory, be used for all values of ; however, there may be severe cancellation when is not large compared with .
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►For evaluation of the Hankel functions and for complex values of and based on the integral representations (10.9.18) see Remenets (1973).
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►Suppose, for example, , and .
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►Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of graphs in (e.
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6: 26.6 Other Lattice Path Numbers
§26.6 Other Lattice Path Numbers
… ►Delannoy Number
… ►Motzkin Number
… ►Narayana Number
… ►§26.6(iv) Identities
…7: 26.11 Integer Partitions: Compositions
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►For example, there are eight compositions of 4: , and .
denotes the number of compositions of , and is the number of compositions into exactly
parts.
is the number of compositions of with no 1’s, where again .
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►The Fibonacci numbers are determined recursively by
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►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).