Explicit algebraic relations between the quantum integrals of the elliptic Calogero–Moser quantum problems related to the root systems A2 and B2 are found.

Two inequalities for weighted sums of powers are established. Applications to Jacobian elliptic functions and Legendre’s elliptic integrals of the first kind are presented. Some known and new inequalities for circular and hyperbolic functions are obtained. Applications to certain iterative means including Gauss’ arithmetic-geometric mean and Schwab-Borchardt mean are included. Mathematics subje...

General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the “classical” special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on author’s habilitation thesis [Spi7] containing a more detailed account of the subject.

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BCn and An; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BCn, we also consider their “Type II” integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint inte...

The classical dynamics of a free particle constrained to move in an integrable two-dimensional confocal elliptic billiard is investigated. We derive the characteristic equations for periodic orbits, classify the orbits, present the Poincaré maps, give expressions for the lengths of the trajectories, and do a stability analysis of special orbits. We also explore some interesting geometrical cons...

We calculate master integrals for bipartite cuts of the three-loop propagator QED diagrams. These determine spectral density photon self energy. Our results are expressed in terms iterated integrals, which, apart from $4m$ cut, reduce to Goncharov's polylogarithms. The cut have been calculated our previous paper one-fold harmonic polylogarithms and complete elliptic integrals. provide threshold...