An interesting $q$-series related to the 4-th symmetrized rank function

Abstract

This paper presents the methods to utilizing the $s$-fold extension of Bailey's lemma to obtain $spt$-type functions related to the symmetrized rank function ${\eta }_{2k}(n).$ We provide the $k=2$ example, but clearly illustrate how deep connections between higher-order spt functions exist for any integer $k>1,$ and provide several directions for possible research. In particular, we present why the function $sp{t}_M^{\ast }(n),$ the total number of appearances of the smallest parts of partitions where parts greater than the smallest plus $M$ do not occur, is an $spt$ function that appears to have central importance.

arXiv commentAdded to the exposition and corrected some typos

References in corpus (5)

• Partition statistics and quasiweak Maass forms
• Bailey Type Transforms and Applications
• Another Smallest Part Function related to Andrews' spt function
• A Strange Partition Theorem Related to the Second Atkin-Garvan Moment
• Some implications of the $2$-fold Bailey lemma

Cited by in corpus (2)

• On an spt function for the $4$-th symmetrized crank function
• A partition identity connected to the second rank moment