sci.math #144246 (0 + 1059 more) ( )--(1)--(1)--[2]
From: wgd@zurich.ai.mit.edu (Bill Dubuque)
Newsgroups: sci.math
[2] Bernoulli numbers everywhere: Umbral Calculus [was: Re: n:th Bernoulli
+ number]
Date: 27 Jun 96 18:53:53
Organization: M.I.T. Artificial Intelligence Lab.
Lines: 1037
Message-ID:
References: <4qf4nn$2e2@mn5.swip.net> <31CF4538.3718@gm.gamemaster.qc.ca>
+
NNTP-Posting-Host: berne.ai.mit.edu
In-reply-to: bruck@pacificnet.net's message of Tue, 25 Jun 1996 18:51:52 -0700
Digressing somewhat, it is surprising just how often Bernoulli numbers
and polynomials arise in diverse contexts. For example, Lang sketches
how they arise in topology and algebraic geometry around Riemann-Roch
theorems, and in analytic and algebraic number theory around zeta
functions and modular forms (see the thread of exercises beginning
with #21 p. 217, end of Chap. IV in Lang's Algebra, 3rd Ed.) One way
of understanding this ubiquity comes from the viewpoint of Hopf
algebras and coalgebras, or, equivalently, the Umbral Calculus. The
latter provides a calculus of adjoints that serves to sytematically
derive and classify almost all of the classical combinatorial
identities for polynomial sequences (e.g the sequences of Abel, Appel,
Bell, Bernoulli, Bessel, Boole, Boas-Buck, Euler, Gould, Hermite,
Laguerre, Mahler, Meixner, Mittag-Leffler, Mott, Poisson-Charlier,
Sheffer, Stirling, etc), along with associated identies (generating
functions, expansions, duplication formulas, recurrences. inversions,
Rodrigues formula, etc, e.g. the Euler-Maclaurin expansion, Boole's
Summation formula, Newton interpolation, Gregory integration,
Vandermonde convolution). For example almost all of the identities
in Riordan's classic book "Combinatorial Identities" can be
systematically derived and classified via the Umbral Calculus. This is
a powerful and useful theory that deserves to be better known. Since
there are now good expositions available. there's no longer any good
reason not to have a peek. I recommend starting with the following
online survey [included in ascii below those Web challenged].
http://www.win.tue.nl/win/math/bs/statistics/bucchianico/hypersurvey/hypersurvey.html
See also the RotaFest page at
http://www-math.mit.edu/~loeb/rotafest.html
-Bill
Note: math formulas are missing from this ascii version, see the URL
above for them.
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