A working seminar on hypergeometric series (HGS)

· When and where? Every Wednesday 3:10-4p.m. at 190 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.

· Previous semester seminar schedules: Fall2007

· Spring 2008 seminar schedule:

Date	Speaker	Title	Abstract
Jan 23, Wed.	Tim Huber	Snakes on a Plane!	Certain polynomials arising in series expansions for zeros of generalized Rogers-Ramanujan functions have interesting symmetry. These polynomials provide a refinement for the enumeration of permutations counted by the coefficients in the Taylor series expansion of Sec z + Tan z about z = 0. Such permutations are called "snakes" or "alternating permutations", and the numbers which enumerate them are called Euler numbers. The polynomials arising from generalized Rogers-Ramanujan series are part of a wider class of polynomials with similar combinatorial interpretations. This is joint work with Ae Ja Yee from Penn. State.
Jan. 30, Wed.	Tim Huber	More snakes on a plane!	Several enumerating functions for alternating permutations will be discussed. The relevant generating functions are connected to geometrically distributed random variables, zeros of generalized Rogers-Ramanujan series, and certain continued fractions.
Feb. 6	Tim Huber	Eisenstein Series	Arithmetic and analytic aspects of Eisenstein series will be discussed. We will review classical results, including relations to hypergeometric series. Applications to certain problems in number theory will be presented as time permits.
Feb. 13	Tim Huber	Eisenstein Series II
Feb. 20	Tim Huber	Eisenstein Series & Hypergeometric Functions
Feb. 27	Tim Huber	Modular Equations from Ramanujan's Notebooks
Mar. 5	Ling long	Zeros of Eisenstein series	In this talk, we will discuss some basic properties of Eisenstein series and in particular the locations of their zeros.
Mar. 12	Jim Wilson	Hypergeometric series
Mar. 26	Jim Wilson	Asymptotic, two techniques
Apr. 2	Jim Wilson	Asymptotic, two techniques II
Apr. 9	Ling Long	Zeros of Eisenstein series II
Apr. 16	Chris Kurth	Transcendentality and the j-function	The j-function is a modular function whose inverse can be written as a quotient of two hypergeometric functions. The Schneider-Lang theorem has to do with when sets of functions simultaneously map a number into a number field. We will use the Schneider-Lang Theorem to prove that if t is algebraic then either t is the solution of a quadratic polynomial or j(t) is transcendental.
Apr. 21, Monday, 4:10-5p.m. in 290 Carver	George Andrews, Penn. State	Old and new thoughts on the Rogers-Ramanujan identities.