1,917 research outputs found
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
Advanced Determinant Calculus: A Complement
This is a complement to my previous article "Advanced Determinant Calculus"
(S\'eminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the
present article, I share with the reader my experience of applying the methods
described in the previous article in order to solve a particular problem from
number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12
(2003), 441-456). Moreover, I add a list of determinant evaluations which I
consider as interesting, which have been found since the appearance of the
previous article, or which I failed to mention there, including several
conjectures and open problems.Comment: AmS-LaTeX, 85 pages; Final, largely revised versio
The two electron molecular bond revisited: from Bohr orbits to two-center orbitals
In this review we first discuss extension of Bohr's 1913 molecular model and
show that it corresponds to the large-D limit of a dimensional scaling
(D-scaling) analysis, as developed by Herschbach and coworkers.
In a separate but synergetic approach to the two-electron problem, we
summarize recent advances in constructing analytical models for describing the
two-electron bond. The emphasis here is not maximally attainable numerical
accuracy, but beyond textbook accuracy as informed by physical insights. We
demonstrate how the interplay of the cusp condition, the asymptotic condition,
the electron-correlation, configuration interaction, and the exact one electron
two-center orbitals, can produce energy results approaching chemical accuracy.
Reviews of more traditional calculational approaches, such as Hartree-Fock, are
also given.
The inclusion of electron correlation via Hylleraas type functions is well
known to be important, but difficult to implement for more than two electrons.
The use of the D-scaled Bohr model offers the tantalizing possibility of
obtaining electron correlation energy in a non-traditional way.Comment: 99 pages, 29 figures, review article, to appear in Advances in
Atomic, Molecular and Optical Physic
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