Shifted and Shiftless Partition Identities II
摘要
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.
引用
@article{arxiv.math/0605317,
title = {Shifted and Shiftless Partition Identities II},
author = {Frank G. Garvan and Hamza Yesilyurt},
journal= {arXiv preprint arXiv:math/0605317},
year = {2007}
}
备注
To appear in Int. J. Number Theory