[Abstract] F.G. Garvan and Hamza Yesilyurt; Shifted and Shiftless Partition Identities II Int. J. Number Theory, 3 (2007), no. 1, 1--42.


Abstract: Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form

p(S,n)=p(T,n-a), for all n > a.
Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M=32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions.

Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form

p(S,n)=p(T,n), for all n is not equal to a.

In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four parameter theta function identity due to Jacobi. New shifted and shiftless partition identities are proved.

The url of this page is http://www.math.ufl.edu/~frank/abstracts/shifted.html.
Created by F.G. Garvan (fgarvan@ufl.edu) on Thursday, May 11, 2006.
Last update made Thu Jan 25 15:08:05 EST 2007.

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