Related papers: Elliptic hypergeometric functions
This is an English translation of Felix Klein's paper "Ueber die Transformation elfter Ordnung der elliptischen Functionen" from 1879.
Recent progress in analytical calculation of the multiple [inverse, binomial, harmonic] sums, related with epsilon-expansion of the hypergeometric function of one variable are discussed.
Finite hypergeometric functions are functions of a finite field ${\bf F}_q$ to ${\bf C}$. They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980's.…
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…
We elaborate on the expansion of hypergeometric functions about rational parameters, where we focus mainly on the integer and half-integer case. The strategy and the basic steps of a recently developed algorithm for the expansion about…
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic…
This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding…
Our purpose in this present paper is to investigate generalized integration formulas containing the extended generalized hypergeometric function and obtained results are expressed in terms of extended hypergeometric function. Certain…
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…
Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely…
In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we…
General elliptic hypergeometric functions are defined by elliptic hypergeometric integrals. They comprise the elliptic beta integral, elliptic analogues of the Euler-Gauss hypergeometric function and Selberg integral, as well as elliptic…
Special functions have always played a central role in physics and in mathematics, arising as solutions of particular differential equations, or integrals, during the study of particular important physical models and theories in Quantum…
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
Li-Chien Shen developed a family of elliptic functions from the hypergeometric function $_2F_1(\frac{1}{3}, \frac{2}{3} ; \frac{1}{2} ; \bullet)$. We comment on this development, offering some new proofs.
Draft version of an article prepared for the Encyclopedia of Mathematical Physics, Elsevier, to appear in 2006.
The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced…
There were some errors in paper hep-th/9303018 in formulas 6.1, 6.6, 6.8, 6.11. These errors have been corrected in the present version of this paper. There are also some minor changes in the introduction.
The first part surveys the push forward formula for elliptic class and various applications obtained in the papers by L.Borisov and the author. In the remaining part we discuss the ring of quasi-Jacobi forms which allow to characterize the…
The parabolic functions are introduced in analogy to the circular and hyperbolic cases. We discuss the relevant properties, the geometrical interpretation and touch on possible generalizations and their link with the modular elliptic…