Many symbolic summations and recurrences that once had to be done with ad hoc methods, special bijections, and the like, can now be done by computer using methods developed or popularized by Wilf and Zeilberger. For example, while a fancy calculator can do a specific numerical computation,
3 ----- \ / 3 \ ) ( ) = 8 / \ k / ----- k = 0
a symbolic algebra system such as maple or mathematica could compute
n n ----- ----- 2 \ / n \ n \ / n \ / 2n \ ) ( ) = 2 and ) ( ) = ( ) / \ k / / \ k / \ n / ----- ----- k = 0 k = 0
while an even fancier system could prove a q-hypergeometric identity such as
n n ----- k ----- k-1 \ q \ (-1) k(k-1)/2 [ n ] ) ------- = ) ------- q [ ] / k / k [ k ] ----- 1 - q ----- 1 - q k = 1 k = 1
There are also extensions to integrals, recurrences, multiple variables, and so on. This is not a completely solved problem, but there are large classes of functions that can be handled systematically.
We will learn the "WZ method" and the various algorithms and mathematics behind these automated computations and proofs. Related topics include holonomic systems; asymptotics; and non-commutative Gröbner bases.
Note: at the present time, the only graduate combinatorics course offerings planned for the 1999-2000 school year are this course (Math 262a; section ID not yet available) and the introductory course Math 264abc (by Jeff Remmel).