On recent Partition function of Kaur and Rana

TL;DR

This paper extends the recently proposed partition function $\rho(n)$, where the largest part $\lambda$ appears exactly once and the remaining parts form a partition of $\lambda$, by deriving generating functions for a broad set of variants, including $\rho_\ell(n)$, $\overline{\rho}(n)$, $\overline{\rho_o}(n)$, $\overline{\rho_e}(n)$, $\overline{\rho}_\ell(n)$, $\rho_{pod}(n)$, $\rho_{ped}(n)$, $\rho_{-k}(n)$, $\rho_c(n)$, and $\rho_{\epsilon}(n)$. The authors present explicit closed forms for these generating functions in terms of $q$-Pochhammer products such as $(q^m;q^m)_\infty$ and simple rational corrections, e.g., $-\frac{1}{1-q^2}$ or similar adjustments. The proofs are elementary, based on standard $q$-series manipulations and decompositions by the largest part together with known generating functions for classical partition statistics like $p(n)$, $b_\ell(n)$, overpartitions, and the $a(n)$ sequence. In addition, they derive a linear recurrence relating $\rho_a(n)$ to $\rho(n)$ via $2\rho_a(n) = n(\rho(n) - 1) + 2 a(n/2)$, where $a(n)$ has generating function $\sum_{n\ge1} a(n) q^n = \frac{1}{(q;q)_\infty}\frac{q}{(1-q)^2}$. The results deepen the understanding of the role of the largest part in restricted partitions and provide compact generating-function tools for analyzing rho-type statistics.

Abstract

Recently, Kaur and Rana introduced the partition function denoted by $ρ(n)$, where the largest part $λ$ appears exactly once, and the remaining parts constitute a partition of $λ$. In this paper, we establish new generating functions for certain variants of $ρ(n)$. Further, we obtain a linear recurrence relation for our new generating function.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• proof : Proof of Theorem \ref{['t:2.1.1']}
• proof : Proof of Theorem \ref{['t:2.1.2']}