A Note on Bi-Orthogonal Polynomials and Functions

Clemente Cesarano

doi:10.3390/fluids5030105

Abstract

The theory of orthogonal polynomials is well established and detailed, covering a wide field of interesting results, as, in particular, for solving certain differential equations. On the other side the concepts and the related formalism of the theory of bi-orthogonal polynomials is less developed and much more limited. By starting from the orthogonality properties satisfied from the ordinary and generalized Hermite polynomials, it is possible to derive a further family (known in literature) of these kind of polynomials, which are bi-orthogonal with their adjoint. This aspect allows us to introduce functions recognized as bi-orthogonal and investigate generalizations of families of orthogonal polynomials

Highlights

The topic of bi-ortogonality will be treated while using the formalism and the operational properties satisfied by different classes of polynomials recognizable as generalized Hermite polynomials [1,2]
The consolidated approach to the study of the characteristics of orthogonality and, less developed, the one related to the concept of bi-ortogonality will be reread on the different formalisms that can be obtained from the various relations deducible from the structure of the different polynomials and the related functions that are attributable to the family of Hermite polynomials [3,4,5]
We will describe the bi-orthogonality relations for a special class of Hermite polynomials, which can be seen as a particular case of the previous definition, but from the other side, as a first step to generalize the concept itself in a different context

Summary

Introduction

The topic of bi-ortogonality will be treated while using the formalism and the operational properties satisfied by different classes of polynomials recognizable as generalized Hermite polynomials [1,2]. The consolidated approach to the study of the characteristics of orthogonality and, less developed, the one related to the concept of bi-ortogonality will be reread on the different formalisms that can be obtained from the various relations deducible from the structure of the different polynomials and the related functions that are attributable to the family of Hermite polynomials [3,4,5]. The fulcrum of the discussion is based on the two-dimensional extension of Hermite polynomials. Hn ( x ) → H(m,n) ( x, y), where we consider the variables as real and the indexes positive integers

Orthogonality and Bi-Orthogonality

Hermite Polynomials and Related Functions

Relevant Relations Involving Bi-Orthogonal Hermite Functions