Abstract

A fractional analogue of classical Gram or discrete Chebyshev polynomials is introduced. Basic properties as well as their relation with the fractional analogue of Legendre polynomials are presented.

1. Introduction

Fractional calculus, in which derivatives, differences, integrals, and sums of fractional order are defined and studied, is nearly as old as the classical calculus of integer order [1]. Its origin goes back to L’Hôpital and Leibniz in 1695 [2] and mathematicians like Fourier, Euler, Laplace, Riemann, and Liouville have made key contributions to it. Fractional calculus has recently been found wide applications in many areas of science and engineering as viscoelastic systems, fluid dynamics, or solid dynamics, to cite some of them [3].

Moreover, orthogonal polynomials and special functions appear in many problems of pure and applied mathematics as, for example, numerical quadrature, least-squares method of approximation, and queueing theory or optics, just to cite a few of them. The systematic analysis of their properties goes back to the XVIII century, in the framework of some problems appearing in celestial mechanics [4]. We refer to [5–9] as basic references on this topic.

Very recently, the problem of numerical evaluation of a fractional integral with unit upper integration limit, has been considered in [10] by using quasi-polynomial of order with commensurate power , giving rise to an orthogonality concept and a new family of orthogonal quasi-polynomials. Moreover, quasi-polynomials orthogonal with respect to fractional densities have been receently introduced in [11].

The main objective of this paper is to introduce a discrete analogue of the quasi-orthogonal polynomials introduced in [10]. In doing so, the paper is structured as follows. In Section 2, we recall the basic definitions and notations. In Section 3, we introduce the fractional Gram orthonormal polynomials. Finally, in Section 4, some numerical tests are presented.

2. Basic Definitions and Notations

Next, we recall some basic facts from the theories of orthogonal polynomials and fractional calculus.

2.1. Classical Legendre and Gram Polynomials

One of the simplest and oldest families of orthogonal polynomials is the Legendre polynomials which can be defined in terms of a Gauss hypergeometric series aswhere denotes the Pochhammer symbol, , and , and satisfy the orthogonality relation: where denotes the Kronecker delta. We will also consider the orthonormal shifted Legendre polynomials, for which the orthogonality relation is over ; that is, Then, the first few shifted Legendre orthonormal polynomials are

Let be a sequence of polynomials orthonormal with respect to a constant weight function at equidistant points: Such a sequence exists [12, 13] and they seem to be the first orthogonal polynomials of a discrete variable introduced in the literature. Properties about their zeros have been obtained in [14]. Nowadays, they are called either the discrete Chebyshev polynomials, in the terminology of [15, 16], or the Gram polynomials [17], in the terminology of [18, 19]. Gram polynomials can be expressed in terms of hypergeometric series asThe following limit relation between Gram polynomials and shifted Legendre orthonormal polynomials holds [7]:

2.2. Fractional Integrals, Derivatives, Sums, and Differences

Next, we recall the definitions of fractional integral, derivative, sum, and difference. We would like to notice here that as compared with the long and rich history of fractional calculus [20–23], discrete fractional calculus attracted the interest of researchers only in a short period of time [24–30].

Definition 1. Let be a given function. The fractional integral of order of is defined as [20, 31]

We would like to notice that for the integral may be singular, but it is well defined if we assume, for example, .

Definition 2. Let . The fractional Riemann-Liouville derivative of order of is defined as [20, 31] provided the right hand side is defined for almost every . The fractional Riemann-Liouville derivative of order is well defined if, for example, is absolutely continuous on every compact interval of .

There are several definitions of fractional integral and fractional derivative [23]. We are not giving a complete list but recall the Caputo fractional derivative [20, 31].

Definition 3. The Caputo fractional derivative is defined as with .

In addition, if is an absolutely continuous function on every compact interval (of ), we can write for

Definition 4 (see [28]). Let ; the fractional sum of of order with base point is defined by where is a function defined for and the falling factorial power function is defined, for , as [32] Thus, is a function defined for .

In particular, maps functions defined on to functions defined on , where .

Let and be the well known forward difference operator of order ; that is,

Definition 5 (see [33], Definition 2.3). Let and , where . The fractional Caputo type difference of order of is defined as for all .

We will use , to denote both differences and sums. More precisely, will denote a fractional sum of order and will be used to denote a fractional difference of order .

2.3. Fractional Orthonormal Legendre Polynomials

A novel class of quasi-polynomials orthogonal with respect to the fractional integration operator has been developed in [10]. The related Gaussian quadrature formulas for numerical evaluation of fractional order integrals have been proposed also in [10]. In doing so, the problem of numerical evaluation of fractional integral with unit upper integration limit (1) has been considered, by using the concept of orthogonality in the following sense: where are defined in (2). We will refer to as fractional Legendre polynomials. The above orthogonality relation can be written as and by using the change of variable , it yields The latter relation can be read in the following sense: is a sequence of polynomials orthogonal with respect to the positive and integrable weight function: From the classical theory of orthogonal polynomials [6, 34], we know that there exists a unique sequence of polynomials orthogonal with respect to , up to a normalizing constant. Since valid for , we have that where is the normalizing constant.

For orthonormal fractional Legendre polynomials, we have Also, where with

From (18), it is obvious that if , we recover orthonormal Legendre polynomials. More precisely, for the first elements, by using their explicit expressions, we have obtained which coincide with (6).

3. Fractional Orthonormal Gram Polynomials

Let be defined as in (2): From (17), for , we define the fractional Gram orthonormal polynomials as the family which satisfies Similar to the fractional Legendre case, we have that is a positive summable weight function, of a discrete variable:

It is clear from the above definition (30), and the following limit relation between fractional orthonormal Gram polynomials and classical orthonormal Gram polynomials defined in (8) holds:

Moreover, by using [5, (5.11.12), p. 141] we have the following limit relation:Hence, we have the following limit relation between fractional orthonormal Gram polynomials and fractional orthonormal Legendre polynomials:which is indeed a fractional extension of (9). In Figure 1, we show the graphs of the first fractional orthonormal Gram polynomials.

Figure 1 

Graphs of the polynomials after the change of variable , for , for the specific values of the parameters , , and .

4. Numerical Experiments

In this section, numerical experiments performed by using Mathematica [35] are presented.

4.1. and

In this case, Therefore, Notice that in accordance with (35).

Moreover, As a consequence, In this case, also in accordance with (35).

4.2. and

For these values of and , we have that Notice that in accordance with (33).

From the above expressions for fractional Gram orthonormal polynomials, we have Notice that in accordance with (35).