English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The least r-gap, $$g_r(\lambda )$$gr(λ), of a partition $$\lambda $$λ is the smallest positive integer that does not appear at least r times as a part of $$\lambda $$λ. In this article, we introduce two new partition functions involving least r-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler’s partition function p(n), polygonal numbers, and the new partition functions. To prove the results, we use an interplay of combinatorial and q-series methods. We also give a combinatorial interpretation for $$\begin{aligned} \sum _{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2). \end{aligned}$$∑n=0∞(±1)k(k+1)/2p(n-r·k(k+1)/2).

Bisected theta series, least r-gaps in partitions, and polygonal numbers