# Density solutions to a class of integro-differential equations

## TL;DR

This paper characterizes the existence and uniqueness of density solutions to a specific class of integro-differential equations, linking them to Beta distributions and extending generalized stable distributions, while also addressing open problems on infinite divisibility.

## Contribution

It provides a complete characterization of density solutions to the equation, expressing them via Beta distributions, and resolves open questions about their infinite divisibility.

## Key findings

- Density solutions exist and are unique if and only if m > α.
- Solutions extend generalized one-sided stable distributions.
- The paper solves open problems on infinite divisibility of these densities.

## Abstract

We consider the integro-differential equation ${\rm I}^{\alpha}_{0+}f= x^m f$ on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if $m> \alpha.$ These density solutions extend the class of generalized one-sided stable distributions introduced in Schneider (1987) and more recently investigated in Pakes (2014). We study various analytical aspects of these densities, and we solve the open problems about infinite divisibility formulated in Pakes (2014).