  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.
