New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions

TL;DR

This paper introduces new weighted partition identities involving the smallest part of partitions, utilizing advanced q-series transformations to extend understanding of partitions, overpartitions, and partitions with distinct even parts.

Contribution

It develops novel weighted partition theorems emphasizing the smallest part, using q-series identities and transformations, expanding the theoretical framework of partition identities.

Findings

Discovered new weighted partition identities involving smallest parts.

Extended partition theory to overpartitions and partitions with distinct even parts.

Utilized q-series transformations like the q-binomial theorem and Jackson's sum.

Abstract

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.