We present our short series of interactive animations on directional derivatives and level curves. This visualisation was developed for students of first-year mathematics courses at the Delft University of Technology. The interactive animation is an animated video that can be interacted with while it is paused or playing. The user can change, for example, the function that is plotted, drag points of interest or change the angle of the camera to explore the scene.

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These lecture notes discuss some of the basics of elliptic hypergeometric functions. These are fairly recent generalizations of ordinary hypergeometric functions. In this chapter we first discuss both ordinary hypergeometric functions and elliptic functions, as you need to know both to define elliptic hypergeometric series. We subsequently discuss some of the important properties these series satisfy, in particular we consider the biorthogonal functions found by Spiridonov and Zhedanov, both with respect to discrete and continuous measure. In doing so we naturally encounter the most important evaluation and transformation formulas for elliptic hypergeometric series, and for the associated elliptic beta integral.@en

Context. 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.

Aims. 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.

Methods. 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 the spectral peaks on the key parameters inclination, obliquity, and cap size is studied.

Results. The ice-scattering and ocean glint contributions can be separated out, because the coefficients for glint are all positive, whereas ice sheets lead to even-numbered, higher harmonics. An in-view polar cap on a planet without axial tilt only produces a single peak. The special situation of edge-on observation, which is important for planets in transit, leads to the most pronounced spectral behavior. Then the respective spectra from planets with a circumventing ocean, a circular ocean (eyeball world), polar caps,
and rings, have characteristic power-law tails  n^-2, n^-7/2, n^-4, and (−1)ⁿ⁺¹ n^-2.

Conclusions. Promising recently discovered planetary systems may be selected as candidates for long-term (multiyear) observation: their Fourier spectra could separate the different planets and reveal or identify a water-covered planet with polar caps.@en

The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the -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.@en