New combinatorial identity for the set of partitions and limit theorems in finite free probability theory

Abstract

We provide a refined combinatorial identity for the set of partitions of $\{1,\dots, n\}$, which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida's limit theorem. That is, we provide the limit of $\{D_{1/m}((p_d^{\boxtimes_d m})^{\boxplus_d m})\}_{m\in \mathbb{N}}$ as $m\rightarrow\infty$ in two cases: (i) $m/d\rightarrow t$ for some $t>0$, or (ii) $m/d\rightarrow0$. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko's limit theorems concerning the unitary Hermite and the Laguerre polynomials.