ABSTRACTDate: September 23, 2010Place: Rutgers Experimental Mathematics Seminar Title: Some new companions to Euler's pentagonal numbers theorem Abstract: We will deduce two new companions to Euler's celebrated Pentagonal numbers theorem from partial theta identiities of Ramanujan and Andrews. These companions are acttually stronger in the sense that they can be refined by the introduction of a parameter $a$. The cancellation by parity split is just as strong as in Euler's theorem even with the parameter $a$, but for these companions the role of the pentagonal numbers is replaced by the squares. It is amazing that even though the weights in the two companions are different, their sums are the same, and so the cancellation and the resulting lacunarity are identical. Finally, from another identity in Ramanujan's Lost Notebook, we will deduce one more companion to Euler's theorem, this one involving twice the pentagonal numbers. We will show how these companions are connected to various fundamental results in the theory of partitions and q-hypergeometric series. The talk will be accessible to non-experts. YouTube version (in 8 parts): Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Part 8
The url of this page is http://krishnaswami-alladi.com/talks/09-23-2010-abstract.html.
alladik@ufl.edu
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