Elliptic Special Functions

California Institute Of Technology, Pasadena CA

Investigators

Abstract

This proposal continues the investigator's research on elliptic analogues of special functions (hypergeometric functions, Painleve transcendents, and Macdonald polynomials), with specific attention to the study of degenerations. One major theme of the research is the classification of difference and differential equations (viewed as difference equations on a possibly very singular elliptic curve) using certain moduli spaces of sheaves on (possibly noncommutative) surfaces, generalizations of the spaces of initial conditions of the Painleve equations. For instance, this reduces the problem of classifying degenerations of the elliptic hypergeometric equation to one of classifying -2 curves on certain rational surfaces. Other problems involve classifying and studying degenerations of the investigator's elliptic analogues of Macdonald/Koornwinder polynomials (with a long-term goal of understanding how the Hecke algebra approach might be generalized); proving special cases for Hall-Littlewood-type polynomials of various multivariate quadratic transformations conjectured by the investigator; and studying limiting cases of a random tiling model with elliptic probabilities, generalizing the uniform distribution on lozenge tilings of a hexagon. Historically, the study of "special functions" originated in the fact that quite a few functions of interest in applications turned out to be members of a single family, the hypergeometric functions; more recently, the "Painleve transcendents" have also begun to play a significant role in applications. Under previous grants, the investigator constructed and studied so-called "elliptic" analogues of these functions (a significant generalization that includes most of the generalized hypergeometric functions and generalied Painleve transcendents in the literature); the current research program continues this investigation, with particular attention to the implications for more classical special functions.