We generalize Warnaar's elliptic extension of a Macdonald multi-parameter summation formula to Riemann surfaces of arbitrary genus. We start by a brief outline of the general theory of hypergeometric type series built from Jacobi theta functions [12, 13] (its extension to integrals [15] is not touched at all). Within this approach univariate elliptic hypergeometric series are defined as the ser...

Motivated by the precision results in the electroweak theory studies of two-loop Feynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop selfenergy diagrams in arbitrary and especially four dimensions. This is done in three respects: firstly results in terms of generalized, multivariable hypergeometric functions are presented giving expl...

Hyperbolic beta integrals are analogues of Euler’s beta integral in which the role of Euler’s gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as (q, q̃ )-bibasic analogues of the beta integral in which the two bases q and q̃ are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under considerat...

The propagation time of a signal, emitted by moving along an elliptical orbit satellite from the GPS (or GLONASS) confi gurations is very important ingredient theory, based on formalism null cone and accounting for effects General Relativity Theory. For case satellites, orbiting plane elliptic orbit, it has been proved that signal between satellites given combination integrals first, second thi...

A bstract Elliptic multiple polylogarithms occur in Feynman integrals and particular scattering amplitudes. They can be characterized by their symbol, a tensor product the so-called symbol letters. In contrast to non-elliptic case, elliptic letters themselves satisfy highly non-trivial identities, which we discuss this paper. Moreover, introduce prime, an analog of for letters, makes these iden...

We provide the monotonicity and convexity properties and sharp bounds for the generalized elliptic integrals [Formula: see text] and [Formula: see text] depending on a parameter [Formula: see text], which contains an earlier result in the particular case [Formula: see text].

We present compact, analytic expressions for the age-redshift relation τ (z) for standard Friedmann-Lemâ itre-Robertson-Walker (FLRW) cosmology. The new expressions are given in terms of Legendre elliptic integrals and evaluate much faster than by direct numerical integration. Subject headings: cosmology: theory – large-scale structure of universe

Complex multiplication in its simplest form is a geometric tiling property. In its advanced form it is a unifying motivation of classical mathematics from elliptic integrals to number theory; and it is still of active interest. This interrelation is explored in an introductory expository fashion with emphasis on a central historical problem, the modular equation between j(z) and j(2z).

Let (P n) be the Catalan-Larcombe-French numbers. The numbers P n occur in the theory of elliptic integrals, and are related to the arithmetic-geometric-mean. In this paper we investigate the properties of the related sequence S n = P n /2 n instead, since S n is an Apéry-like sequence. We prove a congruence and an inequality for P n .

A two-locus genetic model, based on Dawkins "sex war" game, with the fitness of the genotypes at each locus depending on the gene frequencies at the other, is shown to give rise to a stable limit cycle. The mathematical analysis involves averaging techniques and elliptic integrals.