An homotopy of isometries related to a probability density

TL;DR

This paper investigates a family of probability densities linked through homographic relations, deriving isometries from orthogonal polynomial analysis and applying these to solve specific integral equations explicitly.

Contribution

It introduces a novel family of probability densities connected via homographic relations and establishes related isometries using orthogonal polynomial analysis.

Findings

Derived a family of isometries related to probability densities

Connected orthogonal polynomials to integral equation solutions

Provided explicit solutions to specific integral equations

Abstract

We are studying here a family of probability density functions indexed by a real parameter, and constructed from homographic relations between associated Stieltjes transforms. From the analysis of orthogonal polynomials we deduce a family of isometries in relation to the classical operators creating secondary polynomials and we give an application to the explicit resolution of specific integral equations.