Related papers: Elliptic hypergeometric series on root systems

We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…

A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a…

With the use of the $(f,g)$-matrix inversion under specializations that $f=1-xy,g=y-x$, we establish an $(1-xy,y-x)$-expansion formula. When specialized to basic hypergeometric series, this $(1-xy,y-x)$-expansion formula leads us to some…

We lay down the foundations for a systematic study of differentiable and algebraic supervarieties, with a special attention to supergroups.

We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by…

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study…

In this work we derive results concerning Elliptic Functions using as tools general formulas from previus work.

In this note, we shall provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.

A unifying scheme of classical special functions of hypergeometric type obeying orthogonality or biorthogonality relations is described. It expands the Askey scheme of classical orthogonal polynomials and its $q$-analogue based on the…

We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…

In this paper, we establish three new and general transformations with sixteen parameters and bases via Abel's lemma on summation by parts. As applications, we set up a lot of new transformations of basic hypergeometric series. Among…

We obtain extensions of classical hypergeometric identities of Bailey and Whipple that transform nearly-poised and very-well-poised series to Saalsch\"utzian series, Saalsch\"utzian series to Saalsch\"utzian series, and very-well-poised and…

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…

We define elliptic sequences over a commutative ring as sequences indexed by the (positive) integers satisfying a 4-parameter, highly symmetric family of homogeneous quartic relations among terms which we call elliptic relations. We…

We give an alternative proof of an elliptic summation formula of type $BC_n$ by applying the fundamental $BC_n$ invariants to the study of Jackson integrals associated with the summation formula.

Using Krattenthaler's operator method, we give a new proof of Warnaar's recent elliptic extension of Krattenthaler's matrix inversion. Further, using a theta function identity closely related to Warnaar's inversion, we derive summation and…

See Parts I and II in alg-geom/9711032 and alg-geom/9712033. Here we classify maximal hyperbolic root systems of the rank three having restricted arithmetic type and a generalized lattice Weyl vector $\rho$ with $\rho^2<0$ (i. e. of the…

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures…

Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of…