Recurrence and congruences for the smallest parts function

TL;DR

This paper derives recursive formulas and congruences for the smallest parts function in partition theory using Hecke traces and quadratic Dirichlet series, providing new explicit formulas and incongruence results.

Contribution

It introduces generalized Euler-like recursive formulas for the spt function and expresses related power series using Hecke traces, advancing understanding of partition congruences.

Findings

Derived recursive formulas for the spt function.

Expressed power series in terms of Hecke traces.

Established an incongruence result for the spt function.

Abstract

Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet series. As a corollary, we give a closed form expression of the power series $\sum_{n\geq 0}\spt(\ell n-\delta_{\ell})q^n\pmod{\ell}$, $\delta_{\ell}:=(\ell^2-1)/24$, by Hecke traces for weight $\ell+1 $ cusp forms on $\SL_2(\mathbb{Z})$. We further establish an incongruence result for the $\spt$ function.