4,571 research outputs found
Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators
For each of the eight -th derivative parameter changing formulas for Gauss
hypergeometric functions a corresponding fractional integration formula is
given. For both types of formulas the differential or integral operator is
intertwining between two actions of the hypergeometric differential operator
(for two sets of parameters): a so-called transmutation property. This leads to
eight fractional integration formulas and four generalized Stieltjes transform
formulas for each of the six different explicit solutions of the hypergeometric
differential equation, by letting the transforms act on the solutions. By
specialization two Euler type integral representations for each of the six
solutions are obtained
Fast Large Scale Structure Perturbation Theory using 1D FFTs
The usual fluid equations describing the large-scale evolution of mass
density in the universe can be written as local in the density, velocity
divergence, and velocity potential fields. As a result, the perturbative
expansion in small density fluctuations, usually written in terms of
convolutions in Fourier space, can be written as a series of products of these
fields evaluated at the same location in configuration space. Based on this, we
establish a new method to numerically evaluate the 1-loop power spectrum (i.e.,
Fourier transform of the 2-point correlation function) with one-dimensional
Fast Fourier Transforms. This is exact and a few orders of magnitude faster
than previously used numerical approaches. Numerical results of the new method
are in excellent agreement with the standard quadrature integration method.
This fast model evaluation can in principle be extended to higher loop order
where existing codes become painfully slow. Our approach follows by writing
higher order corrections to the 2-point correlation function as, e.g., the
correlation between two second-order fields or the correlation between a linear
and a third-order field. These are then decomposed into products of
correlations of linear fields and derivatives of linear fields. The method can
also be viewed as evaluating three-dimensional Fourier space convolutions using
products in configuration space, which may also be useful in other contexts
where similar integrals appear.Comment: 10+4 pages, published versio
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.
An Integral Equation Involving Legendre Functions
Rodrigues’s formula can be applied also to (1.1) and (1.3) but here the situation is slightly more involved in that the integrals with respect to σ^2 are of fractional order and their inversion requires the knowledge of differentiation and integration of fractional order. In spite of this complication the method has its merits and seems more direct than that employed in [1] and [3]. Moreover, once differentiation and integration of fractional order are used, it seems appropriate to allow a derivative of fractional order with respect to σ^-1 to appear so that the ultraspherical polynomial in (1.3) may be replaced by an (associated) Legendre function. This will be done in the present paper
- …