On the Extension of the Three-Term Recurrence Relation to Probabilities Distributions without Moments

Habib Rebei,Anis Riahi

doi:10.4236/jamp.2018.63051

Abstract

In this paper, we extend the three-term recurrence relation for orthogonal polynomials associated with a probability distribution having a finite moment of all orders to a class of orthogonal functions associated with an infinitely divisible probability distribution µ having a finite moments of order less or equal to four. An explicit expression of these functions will be given in term of the Lévy-Khintchine function of the measure µ.

Highlights

It has been known from [1] and [2] that for every probability distribution μ with finite moments of all orders, there exits a family of monic orthogonal polynomials Pn and a paire of sequences α0,α1, = and w0 1, w1, w2, > 0 satisfying the three-term recurrence relation
Since the random variable with distribution μ can be identified, up to stochastic equivalence ≡, with the position operator q on L2 ( μ ), the previous new formulation of the tri-diagonal Jacobi relation in term of the CAP operators is called the quantum decomposition of the classical random variable
We recall briefly, what has been obtained in the paper [4] around quantum decomposition of random variables with I.D-distributions and having a finite second order moment

Summary

Introduction

Since the random variable with distribution μ can be identified, up to stochastic equivalence ≡ , with the position operator q on L2 ( μ ) , the previous new formulation of the tri-diagonal Jacobi relation in term of the CAP operators is called the quantum decomposition of the classical random variable. This paper is organized as follows: In Section 2, we recall some known facts about the bosonic Fock space and the quantum decomposition of classical random variables without moments, having I.D-distributions, obtained in [12] [4] and [5]. The main result of this paper will be given, so that we compute the action of the position operator q = M x on the orthogonal functions En,α This provide such a generalization of the tri-diagonal recursion relation for OP. The explicit form of theses functions will be given

The Bosonic Fock Space

Notations

Orthogonal Functions and Generalization of the Three-Term Recurrence Relation

Conclusion