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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: 24.10 Arithmetic Properties
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►where , and is an arbitrary integer such that .
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►valid when and , where is a fixed integer.
…valid for fixed integers , and for all and such that .
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►valid for fixed integers , and for all such that
and .
…valid for fixed integers and for all such that .
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4: 33.17 Recurrence Relations and Derivatives
5: 24.16 Generalizations
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►For , Bernoulli and Euler polynomials of order
are defined respectively by
…When they reduce to the Bernoulli and Euler numbers of
order
:
…Also for ,
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►For extensions of to complex values of , , and , and also for uniform asymptotic expansions for large and large , see Temme (1995b) and López and Temme (1999b, 2010b).
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►(This notation is consistent with (24.16.3) when .)
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6: 33.4 Recurrence Relations and Derivatives
7: 33.18 Limiting Forms for Large
8: 24.14 Sums
§24.14 Sums
►§24.14(i) Quadratic Recurrence Relations
… ►§24.14(ii) Higher-Order Recurrence Relations
►In the following two identities, valid for , the sums are taken over all nonnegative integers with . … ►In the next identity, valid for , the sum is taken over all positive integers with . …9: 33.1 Special Notation
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►The main functions treated in this chapter are first the Coulomb radial functions , , (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions , , , (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.
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Curtis (1964a):
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Greene et al. (1979):
nonnegative integers. | |
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, .
, , .
10: 33.13 Complex Variable and Parameters
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►The functions , , and may be extended to noninteger values of by generalizing , and supplementing (33.6.5) by a formula derived from (33.2.8) with expanded via (13.2.42).
►These functions may also be continued analytically to complex values of , , and .
The quantities , , and , given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
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33.13.1
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33.13.2
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