DATE: Tuesday, September 7 (2010), at 1:55pm
PLACE: LIT 368
SPEAKER:
Krishna Alladi
|
TITLE:
|
New companions to Euler's pentagonal numbers theorem
from partial theta identities of Andrews and Ramanujan
|
ABSTRACT:
In the previous lecture we showed that a certain partial theta
identity of Ramanujan yielded a companion to Euler's celebrated pentagonal
numbers theorem. It was also shown that this companion was stronger in
the sense that it could be refined to a weighted partition theorem by the
introduction of a parameter $a$ and that lacunarity was preseved even with
the parameter. We will now deduce another companion to Euler's pentagonal
numbers theorem from a partial theta identity of Andrews. This companion
also has a parameter $a$. It is amazing that even though the weights in the
two companions are different, they sum up to the same values and so the
lacunarity is identical!
Even though this is a sequel to the earlier lecture, this talk will be self contained
and very accessible to non-experts. Also, I will begin this lecture by going through
the concluding portion of the previously lecture to facilitate a smooth transition.
|