Related papers: Spherical Couplings and Multiple Elliptic Integral…
Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated…
Let $E$ be an elliptic curve with complex multiplication by a ring $R$, where $R$ is an order in an imaginary quadratic field or quaternion algebra. We define sesquilinear pairings ($R$-linear in one variable and $R$-conjugate linear in the…
In this paper we shall investigate the problem of the representation of the number of integral points of an elliptic curve modulo a prime number p. We present a way of expressing an exponential sum which involves polynomials of third…
We develop a systematic method for computing the angle combinations at all vertices in an edge-to-edge tiling of the sphere by pentagons with the same five angles. The method is a useful and necessary step in many tiling problems about…
We present some addition theorems for spin-weighted spherical harmonics, generalizing previous results for scalar (spin-zero) spherical harmonics. These addition theorems involve sums over the azimuthal quantum number of products of two…
We give analogs for Euler's constant and ln(4/Pi) of the well-known double integrals for zeta(2) and zeta(3). We also give a series for ln(4/Pi) which reveals it to be an "alternating Euler constant."
We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…
We prove a kind of integral expressions for finite multiple harmonic sums and multiple zeta-star values. Moreover, we introduce a class of multiple integrals, associated with some combinatorial data (called 2-labeled posets). This class…
Spectrum of a certain class of first order conformally invariant operators on the sphere is explicitly computed. The class contains the (elliptic verions of) Rarita-Schwinger operator and its higher spin analogues.
Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson,…
We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.
This paper, pursuing the work started in [10] and [11], holds six new formulae for {\pi}, see equations, through ratios of first kind elliptic integrals and some values of hypergeometric functions of three or four variables of Lauricella…
A solid sphere is considered, with a uniformly distributed infinity of points. Two points being pseudorandomly chosen, the analytical probability density that their separation have a given value is computed, for three types of the…
The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors…
A mathematical model is given for the occurrence of preferred orbits and orbital velocities in a Keplerian system. The result can be extended into energies and other properties of physical systems. The values given by the model fit closely…
Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including…
We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x R^m, and of other manifolds which appear as "nice" hypersurfaces in Euclidean space. The formulas are geometrically…
Since their rediscovery in the 1990s, multiple zeta values have become ubiquitous in many areas of mathematics and physics. Their standard integral and sum representations can usually be traced back to a single source, namely the iterated…
By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.
General structure of the multivariate plain and q-hypergeometric terms and univariate elliptic hypergeometric terms is described. Some explicit examples of the totally elliptic hypergeometric terms leading to multidimensional integrals on…