Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in <i>q</i>-Series

Abstract

For every partition and its conjugation , there is an important invariant , which denotes the number of different parts. That is , . We will derive a series of symmetric q -identities from the invariant in partition conjugation by studying modified Durfee rectangles. The extensive applications of the several symmetric q -identities in q -series [1] will also be discussed. Without too much effort one can obtain much well-known knowledge as well as new formulas by proper substitutions and elementary calculations, such as symmetric identities, mock theta functions, a two-variable reciprocity theorem, identities from Ramanujan’s Lost Notebook and so on.