18,952 research outputs found
Darboux evaluations of algebraic Gauss hypergeometric functions
This paper presents explicit expressions for algebraic Gauss hypergeometric
functions. We consider solutions of hypergeometric equations with the
tetrahedral, octahedral and icosahedral monodromy groups. Conceptually, we
pull-back such a hypergeometric equation onto its Darboux curve so that the
pull-backed equation has a cyclic monodromy group. Minimal degree of the
pull-back coverings is 4, 6 or 12 (for the three monodromy groups,
respectively). In explicit terms, we replace the independent variable by a
rational function of degree 4, 6 or 12, and transform hypergeometric functions
to radical functions.Comment: The list of seed hypergeometric evaluations (in Section 2) reduced by
half; uniqueness claims explained; 34 pages; Kyushu Journal of Mathematics,
201
Geometrical foundations of fractional supersymmetry
A deformed -calculus is developed on the basis of an algebraic structure
involving graded brackets. A number operator and left and right shift operators
are constructed for this algebra, and the whole structure is related to the
algebra of a -deformed boson. The limit of this algebra when is a -th
root of unity is also studied in detail. By means of a chain rule expansion,
the left and right derivatives are identified with the charge and covariant
derivative encountered in ordinary/fractional supersymmetry and this leads
to new results for these operators. A generalized Berezin integral and
fractional superspace measure arise as a natural part of our formalism. When
is a root of unity the algebra is found to have a non-trivial Hopf
structure, extending that associated with the anyonic line. One-dimensional
ordinary/fractional superspace is identified with the braided line when is
a root of unity, so that one-dimensional ordinary/fractional supersymmetry can
be viewed as invariance under translation along this line. In our construction
of fractional supersymmetry the -deformed bosons play a role exactly
analogous to that of the fermions in the familiar supersymmetric case.Comment: 42 pages, LaTeX. To appear in Int. J. Mod. Phys.
A Riemann Hypothesis for characteristic p L-functions
We propose analogs of the classical Generalized Riemann Hypothesis and the
Generalized Simplicity Conjecture for the characteristic p L-series associated
to function fields over a finite field. These analogs are based on the use of
absolute values. Further we use absolute values to give similar reformulations
of the classical conjectures (with, perhaps, finitely many exceptional zeroes).
We show how both sets of conjectures behave in remarkably similar ways.Comment: This is the final version (with new title) as it will appear in the
Journal of Number Theor
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