38,311 research outputs found
Applicability of the -Analogue of Zeilberger's Algorithm
The applicability or terminating condition for the ordinary case of
Zeilberger's algorithm was recently obtained by Abramov. For the -analogue,
the question of whether a bivariate -hypergeometric term has a -pair
remains open. Le has found a solution to this problem when the given bivariate
-hypergeometric term is a rational function in certain powers of . We
solve the problem for the general case by giving a characterization of
bivariate -hypergeometric terms for which the -analogue of Zeilberger's
algorithm terminates. Moreover, we give an algorithm to determine whether a
bivariate -hypergeometric term has a -pair.Comment: 15 page
Special functions from quantum canonical transformations
Quantum canonical transformations are used to derive the integral
representations and Kummer solutions of the confluent hypergeometric and
hypergeometric equations. Integral representations of the solutions of the
non-periodic three body Toda equation are also found. The derivation of these
representations motivate the form of a two-dimensional generalized
hypergeometric equation which contains the non-periodic Toda equation as a
special case and whose solutions may be obtained by quantum canonical
transformation.Comment: LaTeX, 24 pp., Imperial-TP-93-94-5 (revision: two sections added on
the three-body Toda problem and a two-dimensional generalization of the
hypergeometric equation
Coherent Excitation of a Two-Level Atom driven by a far off-resonant Classical Field: Analytical Solutions
We present an analytical treatment of coherent excitation of a Two-Level Atom
driven by a far-off resonant classical field. A class of pulse envelope is
obtained for which this problem is exactly solvable. The solutions are given in
terms of Heun function which is a generalization of the Hypergeometric
function. The degeneracy of Heun to Hypergeometric equation can give all the
exactly solvable pulse shapes of Gauss Hypergeometric form, from the
generalized pulse shape obtained here. We discuss the application of the
results obtained to the generation of XUV.Comment: 9 Pages, 8 Figures. Accepted for Physical Review A as a regular
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