Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 141325, 7 pages
http://dx.doi.org/10.1155/2015/141325

Research Article

On Fractional Orthonormal Polynomials of a Discrete Variable

I. Area,¹ J. D. Djida,² J. Losada,³ and Juan J. Nieto3,4

¹Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade de Vigo, 36310 Vigo, Spain
²African Institute for Mathematical Sciences (AIMS), Limbe Crystal Gardens, P.O. Box 608, Southwest Region, Cameroon
³Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
⁴Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 25 March 2015; Accepted 10 June 2015

Academic Editor: Guang Zhang

Copyright © 2015 I. Area et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 as

where 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 as

The 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).

Furthermore,

in accordance with (33).

By using the above expressions for fractional Gram orthonormal polynomials, we obtain

Notice that

in accordance with (35).

4.3. Interlacing Properties of the Zeros

Let us consider the polynomials obtained from after the change of variable . As it happens in the classical discrete situation [8], we have observed numerically that their zeros are real and simple, they lie in the interval , and there exists at most one zero in each interval , .

As an example, for the specific values of the parameters , , and 100,000, we have obtained the following zeros of the above mentioned polynomials:

5. Conclusions

By using fractional differences, we have introduced fractional Gram orthonormal polynomials. These polynomials converge as to fractional Legendre orthonormal polynomials. Moreover, as , they converge to “classical” Gram orthonormal polynomials. Finally, as and , fractional Gram orthonormal polynomials converge to shifted Legendre orthonormal polynomials. We conjecture, in view of our numerical computations, that the zeros are real and simple and there exists at most one zero inside each of the intervals , .

Conflict of Interests

The authors declare no conflict of interests.

Authors’ Contribution

Each of the authors, I. Area, J. Losada, Juan J. Nieto, and J. D. Djida, contributed to each part of this study equally and read and approved the final version of the paper.

Acknowledgments

The referee deserves special thanks for careful reading of the paper. The work of I. Area has been partially supported by the Ministerio de Economía y Competitividad of Spain under Grant MTM2012–38794–C02–01 and cofinanced by the European Community Fund FEDER. Juan J. Nieto and J. Losada also acknowledge partial financial support by the Ministerio de Economía y Competitividad of Spain under Grants MTM2010–15314 and MTM2013–43014–P and XUNTA under Grant R2014/002 and cofinanced by the European Community Fund FEDER. The first author also thanks the hospitality of the African Institute for Mathematical Sciences Cameroon, AIMS, where the essential part of this research was performed during his visits in November 2014, May 2015, and June 2015.

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