Related papers: Classical elliptic hypergeometric functions and th…
We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric…
We review certain classes of iterated integrals that appear in the computation of Feynman integrals that involve elliptic functions. These functions generalise the well-known class of multiple polylogarithms to elliptic curves and are…
A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical…
We construct a toric generalised K\"ahler structure on $\mathbb{C}P^2$ and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised…
We study monotonicity and convexity properties of functions arising in the theory of elliptic integrals, and in particular in the case of a Schwarz-Christoffel conformal mapping from a half-plane to a trapezoid. We obtain sharp monotonicity…
We consider the Euler type integral associated to the configuration space of points on an elliptic curve, which is an analogue of the hypergeometric function associated to the configuration space of points on a projective line. We calculate…
We have derived some new results for the Mellin transform formulas, as well as for the Gauss hypergeometric function. Also, we have found the connection between the Legendre functions of the second kind. Some of the results obtained we used…
Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the…
We give a new derivation and characterisation of the generalised elliptic genus of Krichever-H\"ohn by means of a functional equation.
The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric…
After briefly reviewing selected Ising and chiral Potts model results, we discuss a number of properties of cyclic hypergeometric functions which appear naturally in the description of the integrable chiral Potts model and its…
Elliptic Macdonald polynomials of sl(2)-type and level 2 are introduced. Suitable limits of elliptic Macdonald polynomials are the standard Macdonald polynomials and conformal blocks. Identities for elliptic Macdonald polynomials, in…
Special matrix functions have recently been investigated for regions of convergence, integral representations and the systems of matrix differential equation that these functions satisfy. In this paper, we find the recursion formulas for…
This survey article (which will appear as a chapter in the book ``Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions'', Springer-Verlag) provides a small collection of basic material on multiple…
We introduce an $\ell$-adic analogue of Gauss's hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ-equation…
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…
The electron repulsion integrals over the Slater-type orbitals with non-integer principal quantum numbers are considered. These integrals are useful in both non-relativistic and relativistic calculations of many-electron systems. They…
In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…
The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here…
We introduce new kind of $p$-adic hypergeometric functions. We show these functions satisfy congruence relations, so they are convergent functions. And we show that there is a transformation formula between our new $p$-adic hypergeometric…