A working seminar on hypergeometric series (HGS)

· When and where? Every Wednesday 2:10-3p.m. at 274 Carver Hall.

· Who? Dr. James Wilson (special functions), Ling Long (number theory), Tim Huber (special functions and number theory),  and YOU!

· What are HGS? http://mathworld.wolfram.com/HypergeometricFunction.html

· Why HGS?  The simple idea of recognizing and classifying series by their term ratios, just as geometric series are recognized by their constant term ratio, leads to tremendous unification of theories and results from combinatorics, physics and virtually all fields where special functions and their identities are important.  Each identity for hypergeometric series admits hundreds of combinatorial and physical interpretations.  A few ad hoc methods, applied to these series, prove to be powerful, just because of the vast applicability of the results.

· Goal: The goal of the working seminar is to study properties of hypergeometric and basic hypergeometric series and selected applications in combinatorics, number theory, differential equations, and Ramanujan's identities.

· Will I be able to understand the talks? Don't worry, we start from the very beginning. Tell us your interests and volunteer to give talks at this seminar

Date	Speaker	Title	Abstract
Sep. 5th	James Wilson	Introduction to HGS
Sep. 12	Ling Long	Mahler measures and HGS
Sep. 19	James Wilson	Some techniques for calculating with hypergeometric series
Sep. 26	James Wilson	Contiguous relations I
Oct. 3	James Wilson	Contiguous relations II
Oct. 10	Tim Huber	Introduction to q-series: Basic tools and Techniques	We will discuss theta functions, q-series, and some elementary techniques used to study these objects. Many of the functions considered will generalize the classical hypergeometric series.
Oct. 17	Tim Huber	The Jacobi-Ramanujan Inversion Formula	Quotients of solutions to hypergeometric differential equations are known to have interesting modular properties. The speaker will give an elementary proof of an identity of this type involving the complete elliptic integral of the first kind. This result first appeared in Jacobi's "Fundamenta Nova" and was re-discovered by Ramanujan.  The Jacobi-Ramanujan inversion formula provides an important connection between hypergeometric series and theta functions.
Oct. 24	Chris Kurth	Hypergeometric Series and the Hyperbolic Disc	We will construct a certain modular form called the lambda function which generates all the modular functions for the group Gamma(2). Next we will look at a quotient of two hypergeometric series whose image tiles a disc with hyperbolic triangles. This quotient turns out to be the inverse of the lambda function.
Oct. 31	Ling Long	Triangular groups and HGS	We will discuss relations between arithmetic triangular groups and hypergeometric series. Potential generalizations of some classical results such as the Jacobi-Ramanujan Inversion Formula will be addressed.
Nov. 7	Tim Huber	Snakes & Zeros of Generalized Rogers-Ramanujan Series	Certain polynomials arising in series expansions for zeros of generalized Rogers-Ramanujan series have interesting symmetry. These polynomials provide a refinement of the enumeration of permutations counted by the coefficients in the Taylor series expansion of Tan z about z = 0. These permutations are called snakes, alternating permutations, zig-zag permutations, or up-down permutations. F. H. Jackson's q-analogue of the tangent function provides a refinement in terms of a statistic known as the inversion number. I am interested in finding a corresponding statistic that provides a combinatorial interpretation for the q-tangent functions and the associated q-polynomials arising in the series expansions for zeros of generalized Rogers-Ramanujan series.
Nov. 13	James Wilson	Introduction to Faber polynomials, and the potential theoretic capacity of a triangle
Nov. 27	James Wilson	Some Analytic Capacities and Faber Polynomials
Dec. 5	James Wilson	Analytic Capacities of Triangles