244 research outputs found
Fourier spectra from exoplanets with polar caps and ocean glint
The weak orbital-phase dependent reflection signal of an exoplanet contains
information on the planet surface, such as the distribution of continents and
oceans on terrestrial planets. This light curve is usually studied in the time
domain, but because the signal from a stationary surface is (quasi)periodic,
analysis of the Fourier series may provide an alternative, complementary
approach.
We study Fourier spectra from reflected light curves for geometrically simple
configurations. Depending on its atmospheric properties, a rotating planet in
the habitable zone could have circular polar ice caps. Tidally locked planets,
on the other hand, may have symmetric circular oceans facing the star. These
cases are interesting because the high-albedo contrast at the sharp edges of
the ice-sheets and the glint from the host star in the ocean may produce
recognizable light curves with orbital periodicity, which could also be
interpreted in the Fourier domain.
We derive a simple general expression for the Fourier coefficients of a
quasiperiodic light curve in terms of the albedo map of a Lambertian planet
surface. Analytic expressions for light curves and their spectra are calculated
for idealized situations, and dependence of spectral peaks on the key
parameters inclination, obliquity, and cap size is studied.Comment: 15 pages, 2 tables, 13 figure
Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions
We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised _10φ_9's and their Nassrallah-Rahman type integral representation
Limits of elliptic hypergeometric biorthogonal functions
The purpose of this article is to bring structure to (basic) hypergeometric
biorthogonal systems, in particular to the q-Askey scheme of basic
hypergeometric orthogonal polynomials. We aim to achieve this by looking at the
limits as p->0 of the elliptic hypergeometric biorthogonal functions from
Spiridonov, with parameters which depend in varying ways on p. As a result we
get 38 systems of biorthogonal functions with for each system at least one
explicit measure for the bilinear form. Amongst these we indeed recover the
q-Askey scheme. Each system consists of (basic hypergeometric) rational
functions or polynomials.Comment: 27 pages. This is a self-contained article which can also be seen as
part 1 of a 3 part series on limits of (multivariate) elliptic hypergeometric
biorthogonal functions and their measure
Limits of multivariate elliptic beta integrals and related bilinear forms
In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p → 0, for given behavior of the parameters as p → 0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We also provide the limits for the discrete version of this bilinear form, which is related to a multivariate extension of the Frenkel-Turaev summation
Properties of generalized univariate hypergeometric functions
Based on Spiridonov's analysis of elliptic generalizations of the Gauss
hypergeometric function, we develop a common framework for 7-parameter families
of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric
functions. In each case we derive the symmetries of the generalized
hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic)
and of type E_6 (trigonometric) using the appropriate versions of the
Nassrallah-Rahman beta integral, and we derive contiguous relations using
fundamental addition formulas for theta and sine functions. The top level
degenerations of the hyperbolic and trigonometric hypergeometric functions are
identified with Ruijsenaars' relativistic hypergeometric function and the
Askey-Wilson function, respectively. We show that the degeneration process
yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the
hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.Comment: 46 page
Limits of multivariate elliptic beta integrals and related bilinear forms
In this article we consider the elliptic Selberg integral, which is a BC_n symmetric multivariate extension of the elliptic beta integral. We categorize the limits that are obtained as p → 0, for given behavior of the parameters as p → 0. This article is therefore the multivariate version of our earlier paper "Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions". The integrand of the elliptic Selberg integral is the measure for the BC_n symmetric biorthogonal functions introduced by the second author, so we also consider the limits of the associated bilinear form. We also provide the limits for the discrete version of this bilinear form, which is related to a multivariate extension of the Frenkel-Turaev summation
Limits of elliptic hypergeometric biorthogonal functions
The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the q-Askey scheme of basic hypergeometric orthogonal polynomials. We aim to achieve this by looking at the limits as p→0 of the elliptic hypergeometric biorthogonal functions from Spiridonov (2003), with parameters which depend in varying ways on p. As a result we get 38 systems of biorthogonal functions with for each system at least one explicit measure for the bilinear form. Amongst these we indeed recover the q-Askey scheme. Each system consists of (basic hypergeometric) rational functions or polynomials
Properties of Generalized Univariate Hypergeometric Functions
Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions
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