A relationship between two old mathematical subjects is observed: the theory of hypergeometric functions and the separability in classical mechanics. Separable potential perturbations of the integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid are expressed through the Appell hypergeometric functions F4 of two variables. Even when the number of degrees of freedom inc...

A very general beam solution of the paraxial wave equation in circular cylindrical coordinates is presented. We call such a field a circular beam (CiB). The complex amplitude of the CiB is described by either the Whittaker functions or the confluent hypergeometric functions and is characterized by three parameters that are complex in the most general situation. The propagation through complex A...

For a finite set A of integral vectors, Gel’fand, Kapranov and Zelevinskii defined a system of differential equations with a parameter vector as a D-module, which system is called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the Disomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the the...

Nonpolynomial basic hypergeometric eigenfunctions of the Askey–Wilson second order difference operator are known to be expressible as very-well-poised 8φ7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new der...

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iterated solutions to the differential equations associated with hypergeometric functions to prove the following result: Theorem 1: The epsilon-expansion of a generalized hypergeome...

We express the zeros of the Weierstass ℘-function in terms of generalized hypergeometric functions. As an application of our main result we prove the transcendence of two specific hypergeometric functions at algebraic arguments in the unit disc. We also give a Saalschützian 4F3–evaluation.

This paper addresses the problem of evaluating P (X > Y ) whereX and Y are independent beta random variables. We cast the problem in terms of a hypergeometric function and use hypergeometric identities to calculate the probability in closed form for certain values of the distribution parameters.

We present calculations for non-planar double-box with four massless/massive external/internal legs/propagators. The results are expressed for arbitrary exponents of propagators and dimension in terms of Lauricella’s hypergeometric functions of three variables and hypergeometric-like multiple series.

We present algorithm qHyper for nding all solutions y(x) of a linear homogeneous q-diierence equation such that y(qx) = r(x)y(x) where r(x) is a rational function of x. Applications include construction of basic hypergeometric series solutions, and deenite q-hypergeometric summation in closed form.