  We use the $q$-binomial theorem, the $q$-Gauss sum, and the ${}_2\phi_1
\rightarrow {}_2\phi_2$ transformation of Jackson to discover and prove many
new weighted partition identities. These identities involve unrestricted
partitions, overpartitions, and partitions with distinct even parts. Smallest
part of the partitions plays an important role in our analysis. This work was
motivated in part by the research of Krishna Alladi.

Number Theory;Combinatorics