2006/05/12 by Frank G. Garvan, Garvan, Frank G., Hamza Yesilyurt +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11P83 #Secondary: 05A17 #math.CO #math.NT #msc:05A17 #msc:11P83
paper · pdf · doi:10.48550/arxiv.math/0605317
To appear in Int. J. Number Theory
arxiv created 2006/05/12 · arxiv updated 2009/12/01
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 greater or equal to 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 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.