### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:**
**Brian Nakamura**, Rutgers University, bnaka {at} math [dot] rutgers [dot] edu
** Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Telescopers for 3D Walks via Residues

Speaker: **Shaoshi Chen**, North Carolina State University

Date: Thursday, October 18, 2012 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Telescopers are linear differential (recurrence) operators satisfied by definite integrals (sums) of certain class of functions (sequences). They are extensively used by combinatorists, such as Wilf, Zeilberger, Gessel etc. to enumerate combinatorial objects through generating functions and to show identities involving integrals or sums of special functions. In this talk, we present a criterion for deciding the existence of telescopers for rational functions which relates to combinatorial problems and an algorithm to construct them if they exist. Our approach is based on the investigation of residues of the input rational functions. We show that the problem of constructing telescopers for rational functions of three variables is equivalent to the problem of constructing telescopers for algebraic functions of two variables. With our method, we will answer some challenging problems on three-dimensional walks. This is a joint work with Manuel Kauers (RISC) and Michael F. Singer (NCSU).

See: http://www.math.rutgers.edu/~bnaka/expmath/