…
►
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).
…
…
►Lozier directed the NIST research, technical, and
support staff associated with the project, administered grants and contracts, together with Boisvert compiled the
Software sections for the Web version of the chapters, conducted editorial and staff meetings, represented the project within NIST and at professional meetings in the United States and abroad, and together with Olver carried out the day-to-day development of the project.
…
►All of the mathematical information contained in the Handbook is also contained in the DLMF, along with additional features such as more graphics, expanded tables, and higher members of some families of formulas; in consequence, in the Handbook there are occasional gaps in the
numbering sequences of equations, tables, and figures.
…
►Among the research, technical, and
support staff at NIST these are B.
…
Zelen.
…
…
►, built-in to the browser)
support for MathML is growing, (see
Browsers supporting MathML).
…
►
Browsers supporting MathML
►The
Firefox browser has traditionally had the strongest
support for MathML and its native MathML is used by default.
►Recent enhancements to the
WebKit engine now provide
support for MathML Core.
…
►Most modern browsers
support ‘Web Fonts’, fonts that are effectively included with a web site.
…
…
►‘✓’ indicates that a software package implements the functions in a section; ‘
a’ indicates available functionality through optional or add-on packages; an empty space indicates no known
support.
…
►
Research Software.
This is software of narrow scope developed as a byproduct
of a research project and subsequently made available at no cost to the public.
The software is often meant to demonstrate new numerical methods or software
engineering strategies which were the subject of a research project. When
developed, the software typically contains capabilities unavailable elsewhere.
While the software may be quite capable, it is typically not professionally
packaged and its use may require some expertise. The software is typically
provided as source code or via a web-based service, and no support is provided.
►
Open Source Collections and Systems.
These are collections of software (e.g. libraries) or interactive systems of
a somewhat broad scope. Contents may be adapted from research software or may
be contributed by project participants who donate their services to the project.
The software is made freely available to the public, typically in source code form.
While formal support of the collection may not be provided by its developers,
within active projects there is often a core group who
donate time to consider bug reports and make updates to the collection.
►
Software Associated with Books.
An increasing number of published books have
included digital media containing software described in the book. Often, the
collection of software covers a fairly broad area. Such software is typically
developed by the book author. While it is not professionally packaged,
it often provides a useful tool for readers to experiment with the concepts
discussed in the book. The software itself is typically not formally supported
by its authors.
►
Commercial Software.
Such software ranges from a collection of reusable software
parts (e.g., a library) to fully functional interactive computing environments
with an associated computing language. Such software is usually professionally
developed, tested, and maintained to high standards. It is available for purchase,
often with accompanying updates and consulting support.
…
§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
.
…
►For other sums involving Bernoulli and Euler
numbers and polynomials see
Hansen (1975, pp. 331–347) and
Prudnikov et al. (1990, pp. 383–386).
§26.7 Set Partitions: Bell Numbers
►
§26.7(i) Definitions
►
is the
number of partitions of
.
…
►
§26.7(ii) Generating Function
…
►For higher approximations to
as
see
de Bruijn (1961, pp. 104–108).
…
►A
permutation with restricted position specifies a subset
.
…The
number of
derangements of
is the
number of permutations with forbidden positions
.
►Let
be the
number of ways of placing
nonattacking rooks on the squares of
.
…
►If
, where no element of
is in the same row or column as any element of
, then
…
►The
number of permutations that avoid
is
…
…
►Equations (
24.5.3) and (
24.5.4) enable
and
to be computed by recurrence.
…
►For algorithms for computing
,
,
, and
see
Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).
►
§24.19(ii) Values of Modulo
►For
number-theoretic applications it is important to compute
for
; in particular to find the
irregular pairs
for which
.
…
►
•
Buhler et al. (1992) uses the expansion
24.19.3
and computes inverses modulo of the left-hand side. Multisectioning
techniques are applied in implementations. See also
Crandall (1996, pp. 116–120).
…