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. ------------------------------------------------------------------------------