A Fuss-type family of positive definite sequences

TL;DR

This paper introduces a two-parameter family of deformed Fuss numbers, providing conditions for their positive definiteness and demonstrating that some associated measures are infinitely divisible under free convolution.

Contribution

It defines a new family of positive definite sequences and explores their infinite divisibility properties in free probability theory.

Findings

Established sufficient conditions for positive definiteness.

Identified measures that are infinitely divisible under free convolution.

Extended Fuss numbers through a novel two-parameter deformation.

Abstract

We study a two-parameter family $a_{n}(p,t)$ of deformations of the Fuss numbers. We show a sufficient condition for positive definiteness of $a_n(p,t)$ and prove that some of the corresponding probability measures are infinitely divisible with respect to the additive free convolution.