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Topic: Bernoulli polynomials

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	ipedia.com: Bernoulli number Article (Site not responding. Last check: 2007-10-18)
		The closed forms are always polynomials in m of degree n+1 and are called Bernoulli polynomials.
		The Bernoulli numbers were first studied by Jakob Bernoulli, after whom they were named by Abraham de Moivre.
		The Bernoulli numbers also appear in the Taylor series expansion of the tangent and hyperbolic tangent functions, in the Euler-Maclaurin formula, and in expressions of certain values of the Riemann zeta function.
	www.ipedia.com /bernoulli_number.html (307 words)

	Bernoulli polynomials -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		Unlike the (Click link for more info and facts about orthogonal polynomials) orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the order of the polynomials goes up.
		In the limit of large order, the Bernoulli polynomials, appropriately scaled, approach the (Click link for more info and facts about sine and cosine functions) sine and cosine functions.
		The Bernoulli and Euler polynomials obey many relations from (Click link for more info and facts about umbral calculus) umbral calculus.
	www.absoluteastronomy.com /encyclopedia/b/be/bernoulli_polynomials.htm (301 words)

	Sums of Consecutive Powers
		To write the expression (2) more concisely, we now define Bernoulli polynomials and explore some of their properties.
		One property of the Bernoulli polynomials is that
		Bernoulli noted that since the Bernoulli polynomials are defined by (3) up to a constant and
	www.math.rutgers.edu /~erowland/sumsofpowers.html (670 words)

	Search Results for polynomial*
		Polynomial approximation was neither discovered nor invented by J L Walsh (which may come as a surprise to some mathematicians).
		Todd polynomials, and certain other closely related polynomials, are much studied today and have played a major role in the study and classification of manifolds.
		A polynomial time method in the length of the input n would be an algorithm which took time bounded by a fixed power of log n (the length of the input).
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=polynomial*&CONTEXT=1 (6586 words)

	Sums of Consecutive Powers Project
		(n) is a polynomial in n of degree 3.
		(n) as a polynomial of degree p + 1, assuming that it is a polynomial of this degree.
		There is a hierarchy of mathematical objects that appear in nature, and the Bernoulli polynomials (and especially the Bernoulli numbers) happen to lie outside of any "direct" construction from the integers.
	www.math.rutgers.edu /~erowland/sumsofpowers-project.html (1333 words)

	Introduction on Bernoulli's numbers
		Bernoulli's numbers play an important and quite mysterious role in mathematics and in various places like analysis, number theory and differential topology.
		Perhaps one of the most important result is Euler-Maclaurin summation formula, where Bernoulli's numbers are contained and which allows to accelerate the computation of slow converging series (see the essay on Euler's constant at [9]).
		According to Louis Saalschültz [17], the term Bernoulli's numbers was used for the first time by Abraham De Moivre (1667-1754) and also by Leonhard Euler (1707-1783) in 1755.
	numbers.computation.free.fr /Constants/Miscellaneous/bernoulli.html (1028 words)

	math lessons - Bernoulli polynomials (Site not responding. Last check: 2007-10-18)
		In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function.
		Unlike the orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the order of the polynomials goes up.
		In the limit of large order, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.
	www.mathdaily.com /lessons/Bernoulli_polynomials (201 words)

	Irreducible Factors And P-Adic Poles Of Higher Order Bernoulli Polynomials - Adelberg (ResearchIndex) (Site not responding. Last check: 2007-10-18)
		We establish the p--adic singularity pattern of the coefficients of the higher order Bernoulli polynomials, and use this to determine all instances of p--Eisenstein behavior.
		5 the Euler and Bernoulli Polynomials (context) - Brillhart - 1969
		4 the Degree of an Irreducible Factor of the Bernoulli Polynom..
	citeseer.ist.psu.edu /adelberg92irreducible.html (387 words)

	Index to On-Line Encyclopedia of Integer Sequences
		Bernoulli numbers, numerators and their factorizations: (2) A027645 A027647 A029762 A029764 A033470 A033474 A035078 A035112 A043295 A043303 A046988 A050925
		Bernoulli numbers, numerators and their factorizations: (3) A053382 A060054 A067778 A068206 A068399 A068528 A069040 A069044 A070192 A070193 A071020 A071772
		Bernoulli numbers, numerators and their factorizations: (4) A073276 A075178 A076547 A076549 A079294 = number of prime factors, A083687 A084217 A085092 A085737 A089170 A089644 A089655
	www.research.att.com /~njas/sequences/Sindx_Be.html (762 words)

	Notes - Broken Time Symmetry
		That's why we want to study more carefully the 'morphism' between the two 'representations': it may illuminate the general principles that underlie the transition from particle to stat mech viewpoints, and in particular, the exact nature of the 'loss' of information in the statistical viewpoint.
		The Bernoulli polynomials have a dual that is quite completely different: some generalized functions.
		So, once we found that the right eigenstates in 3.5.1 were Bernoulli polynomials, then we should have 'known' the result of 3.5.2: the 'dual' of a Bernoulli polynomial is given by the generalized functions that make the Euler-Maclaurin series possible.
	linas.org /theory/time-sym.html (910 words)

	Arnold Adelberg: Publications (Site not responding. Last check: 2007-10-18)
		``Universal Bernoulli polynomials and p-adic congruences,'' Applications of Fibonacci numbers 9 (2002), 1-8.
		``Arithmetic properties of the Nörlund polynomial,'' Discrete mathematics (H. Gould volume) 204 (1999), 5-13.
		Research announcement: ``Irreducible factors and p-adic poles of higher order Bernoulli polynomials,'' C.
	www.math.grin.edu /faculty/Adelberg-publications.html (258 words)

	v5n2
		A discrete inequality of Grüss type in normed linear sapces and applications for the Fourier transform, Mellin transform of sequences, for polynomials with coefficients in normed spaces and for vector valued Lipschitzian mappings, are given.
		The sharpest bound is in terms of the one norm of the Appell polynomial which constitutes the coefficients of the derivative of the function to be approximated.
		In this article, with the help of concept of the harmonic sequence of polynomials, the well known Hermite-Hadamard's inequality for convex functions is generalied to the cases with bounded derivatives of n-th order, including the so-called n-convex functions, from which Hermite-Hadamard's inequality is extended and refined.
	rgmia.vu.edu.au /v5n2.html (952 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		xxi--xxvi); A. Magnus and A. Ronveaux, Laguerre and orthogonal polynomials in 1984 (pp.
		II (pp.\ 247--254); M. Alvarez and G. Sansigre, On polynomials with interlacing zeros (pp.\ 255--258); J. Gilewicz and E. Leopold, On the sharpness of results in the theory of location of zeros of polynomials defined by three-term recurrence relations (pp.
		T. Shamir, Orthogonal polynomials and the partial realization problem (pp.\ 451--458); N. Temme, A class of polynomials related to those of Laguerre (pp.\ 459--464); G. Viano, Numerical inversion of the Laplace transform by the use of Pollaczek polynomials (pp.
	www.math.ucl.ac.be /~magnus/magnus.87f (1117 words)

	Abstract of: Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel and Buchholz ... (Site not responding. Last check: 2007-10-18)
		This is a second paper on finite exact representations of certain polynomials in terms of Hermite polynomials.
		The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials.
		The asymptotic approximations of these polynomials are valid for large values of a certain parameter.
	db.cwi.nl /rapporten/abstract.php?abstractnr=826 (136 words)

	The Frobenius-Perron Operator of the Bernoulli Map (Site not responding. Last check: 2007-10-18)
		which the polynomials cannot move beyond; the radius of the circle of converge is limited by this singularity.
		is a parabola, corresponding to the Bernoulli polynomial
		The topological zeta of the Bernoulli operator can be computed very easily in the polynomial basis because we know the eigenvalues and these form a simple series.
	linas.org /math/chap-gkw/node3.html (2927 words)

	A Simpler Characterization of Sheffer Polynomials
		We illustrate this with three examples: the Hermite polynomials, the Laguerre polynomials and the Bernoulli polynomials of the second kind.
		be the sequence of Bernoulli polynomials of the second kind.
		3, and the resulting coalgebra is in fact a bialgebra with respect to the usual multiplication of polynomials.
	pear.math.pitt.edu /mathzilla/Examples/sheffer.xml (1558 words)

	Bernouilli Numbers
		BN is a library function for computing Bernoulli numbers and polynomials.
		The BERNOULLI NUMBERS command can be used to generate a sequence of Bernoulli numbers.
		is set to the number of Bernoulli numbers to generate.
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/bernnumb.htm (202 words)

	Matches for: Author=filaseta
		Filaseta, Michael On $m$th order Bernoulli polynomials of degree $m$ that are Eisenstein.
		Konyagin, Sergei Squarefree values of polynomials all of whose coefficients are $0$ and $1$.
		Filaseta, Michael Irreducibility criteria for polynomials with nonnegative coefficients.
	www.math.sc.edu /~filaseta/public.html (501 words)

	Higher Order Bernoulli Polynomials And Newton Polygons - Adelberg (ResearchIndex) (Site not responding. Last check: 2007-10-18)
		Abstract: this paper, we are primarily concerned with factorization questions of the Bernoulli polynomials, both over the rational number field Q and over the field of (Update)
		1 Recurrence Sequences and Bernoulli Polynomials of Higher Ord..
		On the degrees of irreducible factors of higher order Bernoulli..
	citeseer.ist.psu.edu /adelberg98higher.html (378 words)

	[No title]
		On page 136 of A First Course in Numerical Analysis by Anthony Ralston and Philip Rabinowitz there is a derivation of the Euler-Maclaurin sum formula.
		They leave out the details and define non-standard Bernoulli polynomials.
		My derivation will fill in the details and use standard Bernoulli polynomials.
	www.getnet.com /~cherry/mathml/em.html (125 words)

	Bernoulli Bibliography: F (Site not responding. Last check: 2007-10-18)
		FIELDS J.C. [1] Related expressions for Bernoulli's and Euler's numbers, Amer.
		[1] Generalizations of the Bernoulli polynomials and numbers and corresponding summation formulas, Bull.
		[1] Properties of Stirling polynomials and a disproof of Robertson's conjecture, J.
	www.mscs.dal.ca /~dilcher/bernf.html (986 words)

	Benjamin Doyon's homepage at Theoretical Physics, Oxford
		Doyon, J. Lepowsky and A. Milas, Twisted modules for vertex operator algebras and Bernoulli polynomials.
		Doyon, Two-point functions of scaling fields in the Dirac theory on the Poincaré disk.
		Doyon, J. Lepowsky and A. Milas, Twisted vertex operators and Bernoulli polynomials.
	www-thphys.physics.ox.ac.uk /users/BenjaminDoyon/pub.html (136 words)

	ON QUANTUM THEORETICAL ORIGINS OF NEWTONIAN TIME
		The matrix elements of φ(n), in the limit of unbounded n, becomes an expansion of the Fourier kernel exp(iqp/Ä§) in Hermite polynomials.
		For that calculation, the roots of Hermite polynomials are needed.
		(iz) is an analytic continuation, by rotation of the complex plane, of Bernoulli polynomials.
	graham.main.nc.us /~bhammel/PHYS/newtqtime.html (15407 words)

	Bernoulli Bibliography: Added in 2000
		WANG YUN KUI, MA WU YU [1] Necessary and sufficient conditions for Bernoulli's numbers and discriminant prime numbers.
		[2] Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials, J.
		[1] Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions.
	www.mscs.dal.ca /~dilcher/bernnew00.html (996 words)

	PlanetMath: Topics on polynomials
		Characteristic polynomial in linear algebra (see also Cayley-Hamilton theorem)
		Anyone with an account can edit this entry.
		This is version 9 of Topics on polynomials, born on 2005-06-19, modified 2005-10-17.
	planetmath.org /encyclopedia/TopicsOnPolynomials.html (69 words)

	A generalization of the Bernoulli polynomials, Pierpaolo Natalini, Angela Bernardini (Site not responding. Last check: 2007-10-18)
		A generalization of the Bernoulli polynomials, Pierpaolo Natalini, Angela Bernardini
		A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions.
		Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.jam/1049725735 (136 words)

	Abstract from Pacific Journal of Mathematics - 208-1-4 - Kwang-Wu Chen (Site not responding. Last check: 2007-10-18)
		In this paper, we investigate the zeta function \begin{align} Z(P,\chi,a,s)&=\sum^\infty_{n_1=1}\cdots\sum^\infty_{n_r=1} \chi_1(n_1)\cdots\chi_r(n_r) \\ &\qquad\cdot P(n_1+a_1,\ldots,n_r+a_r)^{-s}, \end{align} where $a_i\geq 0$, $\chi_i$ is a Dirichlet character with conductor $N_i$, and $P$ is a polynomial satisfying certain conditions.
		Its special values at nonpositive integers are closely related to generalized Bernoulli polynomials.
		Using this fact we can easily get sums of products of Euler polynomials and generalized Bernoulli polynomials.
	nyjm.albany.edu:8000 /PacJ/2003/208-1-4nf.htm (75 words)

	Bernoulli Polynomials
		Compute the Bernoulli number or the Bernoulli polynomial.
		The Bernoulli numbers can be defined by the recurrence relation:
		The Bernoulli polynomials can be defined in terms of the Bernoulli numbers:
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/bn.htm (176 words)

	[No title]
		We have used the fact that the Bernoulli numbers B
		1/2, and the representation of Bernoulli polynomials in terms of Bernoulli numbers.
		Notice that this famous formula can be obtained independantly of the functional equation (from Fourier series of Bernoulli polynomials for example) and permits with (6) to check that the functional equation is fulfilled for even positive integers values of s.
	numbers.computation.free.fr /Constants/Miscellaneous/zetageneralities.html (1521 words)

	Representation formulas for entire functions of exponential type and generalized Bernoulli polynomials (Site not responding. Last check: 2007-10-18)
		We introduce a sequence of polynomials which are extensions of the classic Bernoulli polynomials.
		This generalization is obtained by using the Bessel functions of the first kind.
		We use these polynomials to evaluate explicitly a general class of series containing an entire function of exponential type.
	www.austms.org.au /Publ/Jamsa/V64P3/abs/f66 (76 words)

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