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arXiv · 2511.05857

On an spt function for the $4$-th symmetrized crank function

Alexander E. Patkowski
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Abstract

In this paper we find the smallest part function related to the $4$-th symmetrized crank function, corresponding to the one obtained in Patkowski [11] for the $4$-th symmetrized rank function. This provides us with a direct relationship with Garvan's second order smallest part function. We obtain some congruences for these $spt$ functions, as well as asymptotics and inequalities. A finite $q$-series which generates an $spt$ function related to the second crank moment is also stated. This identity has an important relationship to the one obtained by Patkowski [12] related to the second rank moment.

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BibTeXRIS

Alexander E. Patkowski. 2025-11-08. On an spt function for the $4$-th symmetrized crank function. https://doi.org/10.1016/j.aam.2026.103119

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