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The boundary of a numerical range of a finite matrix is always a nice curve (algebraic, closed and simple), but the equation it satisfies is often very complicated. We will show that, furthermore, there is no hope of describing these curves in terms of distances from the eigenvalues -- as the dimension~2, where the numerical range is just an ellipse, would suggest. But, as we will show, there is a remarkably simple ``coval'' description in terms of distances to \textit{tangent lines}. Provided that one measures these distances not only to the eigenvalues but also to additional points, the most important of which are \textit{secondary values} -- which we will define and describe their algebraic and geometric properties.

We~prove uniqueness of the Moebius invariant semi-inner product on hyperbolic-harmonic functions on the unit ball of the real n-space, i.e. on functions annihilated by the hyperbolic Laplacian on the ball., Comment: 11 pages, no figures

We~describe a Dirichlet-type space of $H$-harmonic functions, i.e. functions annihilated by the hyperbolic Laplacian on~the unit ball of the real $n$-space, as~the analytic continuation (in~the spirit of Rossi and Vergne) of the corresponding weighted Bergman spaces. Characterizations in terms of derivatives are given, and the associated semi-inner product is shown to be Moebius invariant. We~also give a formula for the corresponding reproducing kernel. Our~results solve an open problem addressed by M.~Stoll in his book ``Harmonic and subharmonic function theory on the hyperbolic ball'' (Cambridge University Press, 2016)., Comment: 25 pages, no figures

We are going to study properties of "hypergeometrization" -- an operator which act on analytic functions near the origin by inserting two Pochhammer symbols into their Taylor series. In essence, this operator maps elementary function into hypergeometric. The main goal is to produce number of "change of variable" formulas for this operator which, in turn, can be used to derive great number of transform for multivariate hypergeometric functions., Comment: 26 pages

We analyze the asymptotic distribution of roots of Charlier polynomials with negative parameter depending linearly on the index. The roots cluster on curves in the complex plane. We determine implicit equations for these curves and deduce the limiting density of the root distribution supported on these curves. The proof is based on a determination of the limiting Cauchy transform in a specific region and a careful application of the saddle point method. The obtained result represents a solvable example of a more general open problem., Comment: 26 pages

Using the technique of pedal coordinates we investigate the orbits of a free double linkage. We provide a geometrical construction for them and also show a surprising connection between this mechanical system and orbits around a Black Hole and solutions of Dark Kepler problem.

It was shown that pedal coordinates provides natural framework in which to study force problems of classical mechanics in the plane. A trajectory of a test particle under the influence of central and Lorentz-like forces can be translated into pedal coordinates at once without the need of solving any differential equation. We will generalize this result to cover more general force laws and also show an advantage of pedal coordinates in certain variational problems. These will enable us to link together many dynamical systems as well as problems of calculus of variation. Finally -- as an illustrative example -- we will apply obtained results to compute orbits of Solar sail and Dipole drive.

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence of the weak limit of corresponding root-counting measures and deduce formulas for the supporting curves and density. The obtained result represents a solvable example of a more general problem which is still open. Numerical illustrations of the main result are also involved., Comment: published version

We introduce a natural method of computing antiderivatives of a large class of functions which stems from the observation that the series expansion of an antiderivative differs from the series expansion of the corresponding integrand by just two Pochhammer symbols. All antiderivatives are thus, in a sense, "hypergeometric". And hypergeometric functions are therefore the most natural functions to integrate. This paper would like to make two points: First, the method presented is easy. So much so that it can be taught in undergraduate university level. And second: It may be used to prove some of the more challenging examples computed only by heuristic processes like Method of brackets., Comment: 26 pages

We provide a simple derivation of the corrections for Schwarzschild and Schwarzschild-Tangherlini black hole entropy without knowing the details of quantum gravity. We will follow Bekenstein, Wheeler and Jaynes ideas, using summations techniques without calculus approximations, to directly find logarithmic corrections to well-known entropy formula for black holes. Our approach is free from pathological behaviour giving negative entropy for small values of black hole mass $M$. With the aid of Universality principle we will argue that this purely classical approach could open a window for exploring properties of quantum gravity., Comment: arXiv admin note: text overlap with arXiv:0709.1812, arXiv:0906.4194 by other authors

We will make the case that \textit{pedal coordinates} (instead of polar or Cartesian coordinates) are more natural settings in which to study force problems of classical mechanics in the plane. We will show that the trajectory of a test particle under the influence of central and Lorentz-like forces can be translated into pedal coordinates at once without the need of solving any differential equation. This will allow us to generalize Newton theorem of revolving orbits to include nonlocal transforms of curves. Finally, we apply developed methods to solve the "dark Kepler problem", i.e. central force problem where in addition to the central body, gravitational influences of dark matter and dark energy are assumed., Comment: 23 pages