A matrix approach for the semiclassical and coherent orthogonal polynomials

Abstract

We obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials, and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals.

Introduction

Let us consider a linear functional $\mathcal{U}:\mathbb{P}\to \mathbb{C}$ defined on the linear space $\mathbb{P}$ of polynomials with complex coefficients. A sequence of monic polynomials ${\left\{P_n(x)\right\}}_{n⩾0}$ such that

$$\mathrm{deg}(P_n(x))=n\mathrm{and}〈\mathcal{U}\text{,}{P}_n(x)P_m(x)〉=k_n{\delta }_{n\text{,}m}\mathrm{with}{k}_n\ne 0\text{,}n\text{,}m⩾0\text{,}$$

is said to be the sequence of monic orthogonal polynomials (SMOP) associated with $\mathcal{U}$. The existence of a SMOP can be characterized in terms of the infinite Hankel matrix $H={[u_{i+j}]}_{i\text{,}i⩾0}$, where $u_n=〈\mathcal{U}\text{,}{x}^n〉\text{,}n⩾0$, are called the moments associated with $\mathcal{U}$. Indeed, ${\left\{P_n(x)\right\}}_{n⩾0}$ exists if and only if the leading principal submatrices

$$H_n={[u_{i+j}]}_{i\text{,}j=0}^n\text{,}n⩾0\text{,}$$

of H are nonsingular. In this situation, $\mathcal{U}$ is said to be a quasi-definite or regular [3]. On the other hand, if for every $n⩾0\text{,}\det H_n>0\text{,}\mathcal{U}$ is said to be positive definite and it has the integral representation

$$〈\mathcal{U}\text{,}q(x)〉={\int }_{E}q(x)d\mu (x)\text{,}$$

where $\mu$ is a nontrivial positive Borel measure supported on some infinite subset $E\subset \mathbb{R}$. Assuming $u_0=1$, the most familiar sequences of orthogonal polynomials are the so-called classical families: Jacobi, Laguerre and Hermite polynomials. They correspond to the cases when E has bounded support ($E=[-1\text{,}1]$), E is the positive real axis, and $E=\mathbb{R}$, respectively, and the corresponding probability measures are the Beta, Gamma and normal distributions. There are several ways to characterize the classical orthogonal polynomials: as polynomial solutions of a hypergeometric differential equation, as polynomials expressed by a Rodrigues formula, and as the only sequences of orthogonal polynomials whose derivatives also constitute an orthogonal family.

One of the most important properties of orthogonal polynomials is that they satisfy the three term recurrence relation

$${\mathit{xP}}_n(x)=P_{n+1}(x)+b_n{P}_n(x)+a_n{P}_{n-1}(x)\text{,}n⩾0\text{,}$$

where $P_{-1}(x)≔0\text{,}{b}_n\in \mathbb{R}\text{,}n⩾0$, and $a_n\ne 0\text{,}n⩾1$. If $\mathcal{U}$ is positive definite, we have $a_n>0\text{,}n⩾1$. In a matrix form,

$$\mathit{xP}(x)=\mathit{JP}(x)\text{,}$$

where $P(x)={[P_0(x)\text{,}{P}_1(x)\text{,}\dots ]}^T$ and J is the tridiagonal infinite matrix

$$J=\left(\begin{array}{cccc}\hfill b_0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill a_1\hfill & \hfill b_1\hfill & \hfill 1\hfill & \hfill \ddots \hfill \\ \hfill 0\hfill & \hfill a_2\hfill & \hfill b_2\hfill & \hfill \ddots \hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \end{array}\right)\text{,}$$

called the monic Jacobi matrix associated with ${\left\{P_n(x)\right\}}_{n⩾0}$. It is straightforward to see that the zeros of $P_n$ are the eigenvalues of $J_n$, the $n\times n$ principal leading submatrix of J. On the other hand, given arbitrary sequences ${\{b_n\}}_{n⩾0}$ and ${\{a_n\}}_{n⩾1}$, with $b_n\in \mathbb{R}$ and $a_n\ne 0$, you can define J as in (2) and construct ${\{P_n(x)\}}_{n⩾0}$ by using (1). Then, ${\{P_n(x)\}}_{n⩾0}$ is orthogonal with respect to some linear functional $\mathcal{U}$. This relevant fact is known in the literature as Favard’s theorem (see [3]).

Recently, in [20], a matrix characterization for classical orthogonal polynomials was introduced. Let write

$$P_n(x)={\displaystyle \sum _{j=0}^n}{a}_{n\text{,}j}{x}^j\text{,}n⩾0\text{,}$$

and let define the infinite matrix A with entries $a_{n\text{,}j}$, for $0⩽j⩽n\text{,}n⩾0$, and zero otherwise. Notice that A is a lower triangular matrix whose nth row contains the coefficients of the nth degree orthogonal polynomial with respect to the canonical basis ${\{x^n\}}_{n⩾0}$. Furthermore, since $P_n$ is monic, the diagonal entries are $a_{n\text{,}n}=1$ and, therefore, A is nonsingular. We say that A is the matrix associated with the sequence ${\{P_n(x)\}}_{n⩾0}$. If the polynomials are classical, we will say that A is classical.

Following the notation used in [20], we say that a matrix B is a lower semi-matrix if there exists an integer m such that $b_{i\text{,}j}=0$ whenever $i-j<m$. The entry $b_{i\text{,}j}$ is in the $m$th diagonal $i-j=m$. If B is non zero, we say that B has index m, ind$(B)=m$, if m is the minimum integer such that B has at least one nonzero entry in the $m$th diagonal, also if all the entries in its diagonal of index m are equal to $1\text{,}B$ is called monic. Finally, B is said to be $(n\text{,}m)-$banded if there exists a pair of integers $(n\text{,}m)$ with $n⩽m$ and all the nonzero entries of B lie between the diagonals of indices n and m. It is easy to see that the set of banded matrices is closed under addition and multiplication, despite the fact that the inverse of a banded matrix might not be banded.

Let define the matrices

$$D=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \end{array}\right]\text{,}\hat{D}=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1/2\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1/3\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \end{array}\right]\text{,}X=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \ddots \hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \end{array}\right]\text{,}$$

then we get the following matrix characterization for the orthogonality of a sequence of polynomials.

Theorem 1

Let ${\{P_n(x)\}}_{n⩾0}$ be a monic polynomial sequence and let A be its associated matrix. Then, the sequence ${\{P_n(x)\}}_{n⩾0}$ is orthogonal with respect to some linear functional if and only if $J={\mathit{AXA}}^{-1}$ is a $(-1\text{,}1)$-banded matrix whose entries in the diagonals of indices 1 and −1 are all nonzero.

The proof can be found in [20]. Notice that this is a matrix version of the Favard’s theorem, and the entries of J, i.e., the coefficients of the recurrence relation for the orthogonal polynomials, can be obtained from the matrix A. On the other hand, AD has index 1 and its kth row is the vector $[a_{k\text{,}1}\text{,}2a_{k\text{,}2}\text{,}3a_{k\text{,}3}\text{,}\dots \text{,}{\mathit{ka}}_{k\text{,}k}\text{,}0\text{,}\dots ]$, which corresponds to the derivative of $P_k(x)$. Therefore the matrix $\tilde{A}=\hat{D}\mathit{AD}$ is a monic matrix of index zero and it is associated with the sequence ${\{P_n^{[1]}(x)\}}_{n⩾0}$, where $P_n^{[1]}(x)=P_{n+1}^{\prime }(x)/(n+1)$. Using the fact that a sequence of orthogonal polynomials is classical if and only if the sequence of their derivatives is also orthogonal, the following matrix characterization for classical polynomials is also given in [20].

Theorem 2

Let A be the matrix associated with ${\{P_n(x)\}}_{n⩾0}$. Then A is classical if and only if $A{\tilde{A}}^{-1}$ is a (0, 2)-banded monic matrix.