Variations on the binomial theorem and what one can get from them
by Richard ASKEY (U.Wisconsin-Madison)
Abstract:
The binomial theorem can be generalized in a number of
different ways. One way is to focus on the combinatorial
interpretation of lattice point steps. The binomial coefficient
C(n,k) is the number of paths from (0,0) to (k,n-k) which only move
right and up by one step at a time. This is a refinement of the
fact that the total number of paths of this type of length n is 2^n.
There is a further refinement which can be given using the area
under each path from (0,0) to (k,n-k).
There is a related result which comes up when looking at an
extension of the evaluation of the beta function in terms of the
gamma function, and the idea goes back to Fermat and Wallis when
they were finding the value of the integral of t^k. Ramanujan has
some other extensions, including a bilateral sum which generalizes
the binomial theorem and includes Jacobi's triple product for the
theta function and Jacobi's Fourier series expansions of Jacobi
elliptic functions. Much of this will be described.
Don't be put off by the words you may not know much if anything
about such as "theta function" and "elliptc functions".
No knowledge of them will be needed to understand this talk.