Some New Connection Relations Related to Classical Orthogonal Polynomials

Abstract

In this paper, we deal with a problem of positivity of linear functionals in the linear space of polynomials in one variable with complex coefficients. Some new results of connection relations between the corresponding sequences of monic orthogonal polynomials of classical character are established.

1. Introduction

The criterion of positivity is important in special functions theory, such as approximation theory, moment problem, and orthogonal polynomials and applications, among others, because the solutions of many problems depend on the determination of when a specific function is positive or nonnegative. For example, several authors have focused on the study of linear positive operators and their applications in problems and inequalities. Here, we can mention Aloui [1, 2], Bennett [3], and others. Also, we quote the work of Chen et al. [4] in the total positivity of Riordan arrays.

The linear functional is positive definite if and only if , for every integer , where is the Hankel determinants of order of (see [5]). In order to provide a construction process of positive-definite linear functional from positive-definite linear data, Sfaxi has proved a new construction process and he also gave an example a second-order positive-definite linear functional from a Laguerre positive-definite linear functional [6].

The outline of this paper is as follows. In Section 2, we introduce the basic background and notations to be used throughout the paper. In Section 3, we present our main results. Indeed, using Oppenheim’s inequality we prove that, for any pair and any positive-definite linear functional , the linear functional given by is also positive definite. Finally, some new results of connection relations between the corresponding sequences of monic classical orthogonal polynomials are presented.

2. Orthogonality and Positive-Definite Linear Functional Character

Let be the linear space of polynomials in one variable with complex coefficients and its algebraic dual space. We denote by the action of on and by , , the sequence of moments of with respect to the polynomial sequence . Let us define the following operations in . For any linear functionals and , any polynomial , and any , let , , , , and be the linear functionals defined by duality (see [7, 8]):where the right-multiplication of by is a polynomial given by

For any , , where is the Dirac functional defined by .

Notice that if is such that , then .

A linear functional is said to be quasi-definite (regular) if we can associate with it a monic polynomial sequence (MPS) with , such thatwhere is the Kronecker delta.

In such a situation, is said to be the monic orthogonal polynomial sequence (MOPS) with respect to . The MOPS satisfies a three-term recurrence relation (TTRR):

where and for every integer and with . Furthermore, we need the following formula:

When is quasi-definite and such that , then .

A linear functional is said to be positive-definite, if it satisfies (3) with , for every integer .

The quasi-definiteness of a linear functional is equivalent to (see [9, 10])where is the Hankel determinant of order of .

In this way, we have the following formula [5]:

As a consequence, is positive-definite if and only if , for every integer .

Lemma 1. (see [8]). Let be a MOPS with respect to the linear functional and let be a monic polynomial with . The following statements are equivalent:(i)The linear functional is quasi-definite.(ii)There exists a MPS and parameters satisfying

In this case, is the MOPS with respect to .

The Hadamard product of two matrices and is defined by . The Hadamard product arises in a wide variety of ways. It was perhaps the first significant result published about the Hadamard product that the class of positive semidefinite matrices of a given size is closed under the Hadamard product. For more information, the authors may refer to [11]. Oppenheim in [12] proved the following:

Lemma 2. (see [12]). Let and be two positive semidefinite matrices. Then,

3. Main Results

For any with and , let us consider the linear functional where its moments with respect to the MPS that we denote by , are given by

Equivalently, satisfies the distributional equation:

3.1. Preservation of the Positive-Definite Property

Assuming that is positive definite, our purpose is to find the necessary and sufficient condition on the real parameter , in order that the linear functional to be also positive definite.

Theorem 1. Let be a real number with , let , and let be a positive-definite linear functional. The following statements are equivalent:(i).(ii)the linear functional satisfying

Proof. Taking (10) into account, we can always decompose the determinant of Hankel associate to linear functional thanks to Hadamard’s product as follows:Using Lemma 2 (Oppenheim’s inequality) and , for every integer ( be a positive-definite linear functional), we find , if and only if . Hence, the required result is as follows.

3.2. Classical Case

A quasi-definite linear functional is said to be classical if it satisfies a functional equation (Pearson equation):where and are polynomials such that is monic, , and .

The corresponding is said to be classical (for more details, see [8, 13] and the literature therein). The classical character of a linear functional is invariant by shifting. Indeed, if is a classical linear functional satisfying (10), then for any pair with , the shifted linear functional is also classical and satisfies with and , where . We find in [8] a description of the classical linear functional (Hermite, Laguerre, Bessel, and Jacobi).

Proposition 1. If is a classical linear functional with , then

Proof. . So, we can write . Let with . From (14), it follows thatClearly, we have . Accordingly, one has

Now, let us study the particular situation:

Namely, , , , and . The Laguerre linear functional with parameter , which is the unique linear functional satisfying (see [8])

Since and , accordingly we obtain

The MOPS of Laguerre satisfies the first and second structure relations [6, 8]:

Corollary 1. The polynomials and satisfy the following connection relation:

Proof. Using (20) and (21) and by taking (19) into account,

Proposition 2. If is a classical linear functional with , then

Proof. and . Let be a zero of . From the Euclidean division of by , we always havewhere with . Clearly, is given bySince , then is the leading coefficient of .
From (14) and (25), we can writeNotice that , otherwise the linear functional will satisfy and then its moments with respect to the MPS are such that . So, there exists an integer such that , i.e., which contradicts the quasi-definiteness of .
Directly, we have

Now, let us study some particular situations.

(Bessel case): namely, , , , and . The Bessel quasi-definite linear functional with parameter , which is the unique monic linear functional satisfying

We can easily show that

It follows thatso that

Denote by the MOPS with respect to the Bessel linear functional . The MOPS satisfies the first structure relation [8]:where

Corollary 2. The polynomials and satisfy the following relation:

Proof. Using (33), we obtain for any integer where .
By Lemma 1, with , , and , and by taking (22) and (24) into account, the MPS is orthogonal with respect to the linear functional . Thus, , . This leads to (35).
(Jacobi case): namely, , , , and . The Jacobi quasi-definite linear functional with parameters and where and , which is the unique monic linear functional satisfyingSinceit follows thatSo, we can deduce thatDenote by the MOPS with respect to the Jacobi linear functional . The MOPS satisfies the first structure relation for [8]:where

Corollary 3. The polynomials and satisfy the following relation:where for any integer ,

Proof. Using (41), we obtain for any integer ,whereBy Lemma 1, with , , , and by taking (38) and (40) into account, xthe MPS is orthogonal with respect the linear functional . Thus, , . This leads to (43).

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number RGP-2019-5.