# On Fractional Orthonormal Polynomials of a Discrete Variable

Authors: I. Area; J. D. Djida; J. Losada; Juan J. Nieto
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## 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.


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## Body

## 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,(1)1Γα∫01fτ1-τα-1dτ,has been considered in [10] by using quasi-polynomial of order n with commensurate power β, (2)Pntβ=Pn,βt=∑k=0nan,ktkβ,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 polynomialsPn(x) which can be defined in terms of a Gauss hypergeometric series as(3)Pnx=F12-n,n+11∣1-x2=∑k=0n-nkn+1kk!21-x2k,where (A)s denotes the Pochhammer symbol, (A)0=1, and (A)s=A(A+1)⋯(A+s-1), and satisfy the orthogonality relation: (4)∫-11PnxPmxdx=12n+1δn,m,where δn,m denotes the Kronecker delta. We will also consider the orthonormal shifted Legendre polynomials, for which the orthogonality relation is over [0,1]; that is, (5)Rnx=2n+1Pn1-2x=2n+1F12-n,n+11∣x.Then, the first few shifted Legendre orthonormal polynomials are (6)R0x=1,R1x=31-2x,R2x=56x-1x+1.Let{Gn(x;N)} be a sequence of polynomials orthonormal with respect to a constant weight function at N equidistant points: (7)∑x=0NGnx;NGmx;N=δn,m.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(8)Gnx;N=2n+1N-n+1nN+1n+1F23-n,n+1,-x1,-N∣1,=2n+1N-n+1nN+1n+1∑k=0n-nkn+1k-xkk!2-Nk,n=0,1,…,N.The following limit relation between Gram polynomials and shifted Legendre orthonormal polynomials holds [7]: (9)limN→∞⁡NGnNx;N=Rnx.


### 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.
Letf:R→R be a given function. The fractional integral of order α>0 of f is defined as [20, 31] (10)Iαft=1Γα∫0tt-sα-1fsds,t>0.We would like to notice that forα<1 the integral may be singular, but it is well defined if we assume, for example, f∈Lloc1(R).Definition 2.
Letα∈(0,1). The fractional Riemann-Liouville derivative of order α of f is defined as [20, 31] (11)DαftRL=D1I1-αft=1Γ1-αddt∫0tt-s-αfsds,provided the right hand side is defined for almost every t∈R+. The fractional Riemann-Liouville derivative of order α is well defined if, for example, f is absolutely continuous on every compact interval of R.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(12)DαftC=DαgtRL,with g(t)=f(t)-f(0).In addition, iff is an absolutely continuous function on every compact interval (of R), we can write for α∈(0,1)(13)DαftC=I1-αD1ft=1Γ1-α∫0tt-s-αf′sds.Definition 4 (see [28]).
Letα>0; the fractional sum of f of order α with base point a∈Z is defined by (14)Δa-αft=1Γα∑s=at-αt-s-1α-1fs,where f is a function defined for s=amod1 and the falling factorial power function is defined, for α∈R, as [32] (15)tα=Γt+1Γt+1-α.
Thus,Δa-αf is a function defined for t=(a+α)mod1.In particular,Δa-α maps functions defined on Na to functions defined on Na+α, where Nt={t,t+1,t+2,…}.Letn∈N and Δn be the well known forward difference operator of order n; that is, (16)Δnfs=∑k=0n-1n-knkfs+k.Definition 5 (see [33], Definition 2.3).
Letα>0 and n-1<α≤n, where n=α. The fractional Caputo type difference of order α of f is defined as (17)ΔaαftC=Δa-n-αΔnft=1Γn-α∑s=at-n+αt-s-1n-α-1Δnfs,for all t∈Na+α.We will useΔ, to denote both differences and sums. More precisely, Δa-α will denote a fractional sum of order α>0 and Δaα will be used to denote a fractional difference of order α>0.


### 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: (18)1Γα∫01Pn,βτ;αPm,βτ;α1-τα-1dτ=0,n≠m,where Pn,β(τ;α)=Pn(τβ;α) are defined in (2). We will refer to Pn,β(τ;α) as fractional Legendre polynomials. The above orthogonality relation can be written as (19)1Γα∫01Pnτβ;αPmτβ;α1-τα-1dτ=0,n≠m,and by using the change of variable x=τβ, it yields (20)1βΓα∫01Pnx;αPmx;α1-x1/βα-1x1-β/βdx=0,n≠m.The latter relation can be read in the following sense: {Pn(x;α)} is a sequence of polynomials orthogonal with respect to the positive and integrable weight function: (21)wx;α,β=1-x1/βα-1x1-β/β.From the classical theory of orthogonal polynomials [6, 34], we know that there exists a unique sequence of polynomials orthogonal with respect to w(x;α,β), up to a normalizing constant. Since (22)μn=1βΓα∫01xnwx;α,βdx=Γβn+1Γα+βn+1,n=0,1,…,valid for α,β>0, we have that (23)Pnx;α=ϵnμ0μ1μ2⋯μnμ1μ2μ3⋯μn+1⋮⋮⋮⋱⋮μn-1μnμn+1⋯μ2n-11xx2⋯xn,where ϵn is the normalizing constant.For orthonormal fractional Legendre polynomials, we have(24)P0,βτ;α=Γα+1P1,βτ;α=Γα+2β+1Γα+1Γβ+1-τβΓα+β+1Γ2β+1Γ2α+β+1-Γα+1Γ2β+1Γα+2β+1.Also, (25)P2,βτ;α=ϖ1ϖ2τ2βϖ3+τβϖ4+ϖ5,where (26)ϖ2=Γα+1Γα+β+1·Γ22β+1Γα+β+1Γα+3β+1-Γβ+1Γ3β+1Γ2α+2β+1,ϖ3=Γα+2β+1·Γα+1Γ2β+1Γα+2β+1-Γ2β+1Γ2α+β+1Γα+3β+1,ϖ4=Γα+β+1Γα+2β+1·Γ3β+1Γα+β+1Γα+2β+1-βΓα+1ΓβΓ2β+1Γα+3β+1,ϖ5=Γα+1Γα+β+1·Γ22β+1Γα+β+1Γα+3β+1-Γβ+1Γ3β+1Γα+2β+12,ϖ1=Γα+4β+1ϖ6Γα+1Γα+β+1·Γβ+1Γ3β+1Γ2α+2β+1-Γ22β+1Γα+β+1Γα+3β+1,with (27)ϖ6=Γα+1Γ2β+1Γα+2β+1-Γ2β+1Γα+β+12Γ4β+1Γ2α+2β+1·Γα+1Γ2β+1Γα+2β+1-Γ2β+1Γα+β+12×Γ2α+3β+1+Γα+β+1Γα+β+1·Γα+1Γ32β+1Γ2α+3β+1+Γ23β+1Γ3α+2β+1-2Γα+1Γβ+1Γ2β+1Γ3β+1Γ2α+2β+1×Γα+3β+1·Γα+4β+1.From (18), it is obvious that if α,β→1, we recover orthonormal Legendre polynomials. More precisely, for the first elements, by using their explicit expressions, we have obtained (28)limα,β→1⁡P0,βτ;α=1,limα,β→1⁡P1,βτ;α=31-2τ,limα,β→1⁡P2,βτ;α⁡=56τ-1τ+1,which coincide with (6).


## 2.1. Classical Legendre and Gram Polynomials

One of the simplest and oldest families of orthogonal polynomials is the Legendre polynomialsPn(x) which can be defined in terms of a Gauss hypergeometric series as(3)Pnx=F12-n,n+11∣1-x2=∑k=0n-nkn+1kk!21-x2k,where (A)s denotes the Pochhammer symbol, (A)0=1, and (A)s=A(A+1)⋯(A+s-1), and satisfy the orthogonality relation: (4)∫-11PnxPmxdx=12n+1δn,m,where δn,m denotes the Kronecker delta. We will also consider the orthonormal shifted Legendre polynomials, for which the orthogonality relation is over [0,1]; that is, (5)Rnx=2n+1Pn1-2x=2n+1F12-n,n+11∣x.Then, the first few shifted Legendre orthonormal polynomials are (6)R0x=1,R1x=31-2x,R2x=56x-1x+1.Let{Gn(x;N)} be a sequence of polynomials orthonormal with respect to a constant weight function at N equidistant points: (7)∑x=0NGnx;NGmx;N=δn,m.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(8)Gnx;N=2n+1N-n+1nN+1n+1F23-n,n+1,-x1,-N∣1,=2n+1N-n+1nN+1n+1∑k=0n-nkn+1k-xkk!2-Nk,n=0,1,…,N.The following limit relation between Gram polynomials and shifted Legendre orthonormal polynomials holds [7]: (9)limN→∞⁡NGnNx;N=Rnx.


## 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.
Letf:R→R be a given function. The fractional integral of order α>0 of f is defined as [20, 31] (10)Iαft=1Γα∫0tt-sα-1fsds,t>0.We would like to notice that forα<1 the integral may be singular, but it is well defined if we assume, for example, f∈Lloc1(R).Definition 2.
Letα∈(0,1). The fractional Riemann-Liouville derivative of order α of f is defined as [20, 31] (11)DαftRL=D1I1-αft=1Γ1-αddt∫0tt-s-αfsds,provided the right hand side is defined for almost every t∈R+. The fractional Riemann-Liouville derivative of order α is well defined if, for example, f is absolutely continuous on every compact interval of R.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(12)DαftC=DαgtRL,with g(t)=f(t)-f(0).In addition, iff is an absolutely continuous function on every compact interval (of R), we can write for α∈(0,1)(13)DαftC=I1-αD1ft=1Γ1-α∫0tt-s-αf′sds.Definition 4 (see [28]).
Letα>0; the fractional sum of f of order α with base point a∈Z is defined by (14)Δa-αft=1Γα∑s=at-αt-s-1α-1fs,where f is a function defined for s=amod1 and the falling factorial power function is defined, for α∈R, as [32] (15)tα=Γt+1Γt+1-α.
Thus,Δa-αf is a function defined for t=(a+α)mod1.In particular,Δa-α maps functions defined on Na to functions defined on Na+α, where Nt={t,t+1,t+2,…}.Letn∈N and Δn be the well known forward difference operator of order n; that is, (16)Δnfs=∑k=0n-1n-knkfs+k.Definition 5 (see [33], Definition 2.3).
Letα>0 and n-1<α≤n, where n=α. The fractional Caputo type difference of order α of f is defined as (17)ΔaαftC=Δa-n-αΔnft=1Γn-α∑s=at-n+αt-s-1n-α-1Δnfs,for all t∈Na+α.We will useΔ, to denote both differences and sums. More precisely, Δa-α will denote a fractional sum of order α>0 and Δaα will be used to denote a fractional difference of order α>0.


## 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: (18)1Γα∫01Pn,βτ;αPm,βτ;α1-τα-1dτ=0,n≠m,where Pn,β(τ;α)=Pn(τβ;α) are defined in (2). We will refer to Pn,β(τ;α) as fractional Legendre polynomials. The above orthogonality relation can be written as (19)1Γα∫01Pnτβ;αPmτβ;α1-τα-1dτ=0,n≠m,and by using the change of variable x=τβ, it yields (20)1βΓα∫01Pnx;αPmx;α1-x1/βα-1x1-β/βdx=0,n≠m.The latter relation can be read in the following sense: {Pn(x;α)} is a sequence of polynomials orthogonal with respect to the positive and integrable weight function: (21)wx;α,β=1-x1/βα-1x1-β/β.From the classical theory of orthogonal polynomials [6, 34], we know that there exists a unique sequence of polynomials orthogonal with respect to w(x;α,β), up to a normalizing constant. Since (22)μn=1βΓα∫01xnwx;α,βdx=Γβn+1Γα+βn+1,n=0,1,…,valid for α,β>0, we have that (23)Pnx;α=ϵnμ0μ1μ2⋯μnμ1μ2μ3⋯μn+1⋮⋮⋮⋱⋮μn-1μnμn+1⋯μ2n-11xx2⋯xn,where ϵn is the normalizing constant.For orthonormal fractional Legendre polynomials, we have(24)P0,βτ;α=Γα+1P1,βτ;α=Γα+2β+1Γα+1Γβ+1-τβΓα+β+1Γ2β+1Γ2α+β+1-Γα+1Γ2β+1Γα+2β+1.Also, (25)P2,βτ;α=ϖ1ϖ2τ2βϖ3+τβϖ4+ϖ5,where (26)ϖ2=Γα+1Γα+β+1·Γ22β+1Γα+β+1Γα+3β+1-Γβ+1Γ3β+1Γ2α+2β+1,ϖ3=Γα+2β+1·Γα+1Γ2β+1Γα+2β+1-Γ2β+1Γ2α+β+1Γα+3β+1,ϖ4=Γα+β+1Γα+2β+1·Γ3β+1Γα+β+1Γα+2β+1-βΓα+1ΓβΓ2β+1Γα+3β+1,ϖ5=Γα+1Γα+β+1·Γ22β+1Γα+β+1Γα+3β+1-Γβ+1Γ3β+1Γα+2β+12,ϖ1=Γα+4β+1ϖ6Γα+1Γα+β+1·Γβ+1Γ3β+1Γ2α+2β+1-Γ22β+1Γα+β+1Γα+3β+1,with (27)ϖ6=Γα+1Γ2β+1Γα+2β+1-Γ2β+1Γα+β+12Γ4β+1Γ2α+2β+1·Γα+1Γ2β+1Γα+2β+1-Γ2β+1Γα+β+12×Γ2α+3β+1+Γα+β+1Γα+β+1·Γα+1Γ32β+1Γ2α+3β+1+Γ23β+1Γ3α+2β+1-2Γα+1Γβ+1Γ2β+1Γ3β+1Γ2α+2β+1×Γα+3β+1·Γα+4β+1.From (18), it is obvious that if α,β→1, we recover orthonormal Legendre polynomials. More precisely, for the first elements, by using their explicit expressions, we have obtained (28)limα,β→1⁡P0,βτ;α=1,limα,β→1⁡P1,βτ;α=31-2τ,limα,β→1⁡P2,βτ;α⁡=56τ-1τ+1,which coincide with (6).


## 3. Fractional Orthonormal Gram Polynomials

LetGn,β(τ;α,N)=Gn(τβ;α,N) be defined as in (2): (29)Gntβ;α,N=Gn,βt;α,N=∑k=0ngn,kα,Ntkβ.From (17), for N=t-α, we define the fractional Gram orthonormal polynomials as the family which satisfies (30)1Γα∑s=0NN+α-s-1α-1Gnsβ;α,N·Gmsβ;α,N=δn,m,0≤n,m≤N.Similar to the fractional Legendre case, we have that (N+α-s-1)(α-1) is a positive summable weight function, of a discrete variable: (31)∑s=0NN+α-s-1α-1=Γα+1ΓN-α+1/ΓN-2α+1-Γ-α/Γ-2α,0<α<2.It is clear from the above definition (30), and (32)N+α-s-1α-1=Γα+N-sΓ1+N-s,s=0,1,…,N,0<α<1the following limit relation between fractional orthonormal Gram polynomials and classical orthonormal Gram polynomials defined in (8) holds: (33)limα,β→1⁡Gntβ;α,N=Gnt;N,n=0,…,N.Moreover, by using [5, (5.11.12), p. 141] we have the following limit relation:(34)limN→∞⁡N1-αN+α-Ns-1α-1=1-sα-1.Hence, we have the following limit relation between fractional orthonormal Gram polynomials and fractional orthonormal Legendre polynomials:(35)limN→∞⁡Nα/2GnNtβ,α,N=Pn,βt;α,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 polynomialsGn,β(τ;α,N) after the change of variable x=τβ, for n=2,3,4,5, for the specific values of the parameters α=3/5, β=1/7, and N=100,000.


## 4. Numerical Experiments

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


### 4.1.α=3/5 and β=1/7

In this case,(36)G1,βτ;α,1000=-0.430804x7+1.05357,G1,βτ;α,10000=-0.158489x7+0.53835,G1,βτ;α,100000=-0.0573041x7+0.270416.Therefore, (37)Nα/2G1,βNτ;α,N=-9.18017x7+8.36879,N=1000,Nα/2G1,βNτ;α,N=-9.36325x7+8.53227,N=10000,Nα/2G1,βNτ;α,N=-9.3858x7+8.55131,N=100000.Notice that (38)P1,βτ;α=-9.39005x7+8.55459,in accordance with (35).Moreover,(39)G2,βτ;α,1000=0.753333x2/7-3.26517x7+3.42283,G2,βτ;α,10000=0.259692x2/7-1.58875x7+2.3636,G2,βτ;α,100000=0.0700715x2/7-0.596479x7+1.23532.As a consequence, (40)Nα/2G2,βNτ;α,N=43.0656x2/7-69.5788x7+27.1885,N=1000,Nα/2G2,βNτ;α,N=57.1894x2/7-93.8606x7+37.4605,N=10000,Nα/2G2,βNτ;α,N=59.4447x2/7-97.697x7+39.0641,N=100000.In this case, (41)P2,βτ;α=59.7007x2/7-98.1278x7+39.2423,also in accordance with (35).


### 4.2.α=99/100 and β=99/100

For these values ofα and β, we have that (42)G1,βτ;α,1000=0.0570034-0.000120685x99/100,G1,βτ;α,10000=0.0182768-3.9673·10-6x99/100,G1,βτ;α,100000=0.00584794-1.2992·10-7x99/100.Notice that (43)G1x;1000=0.0546902-0.00010938x,G1x;10000=0.0173179-3.46358·10-6x,G1x;100000=0.00547714-1.0954·10-7x,in accordance with (33).From the above expressions for fractional Gram orthonormal polynomials, we have(44)Nα/2G1,βNτ;α,N=1.74141-3.44076x99/100,N=1000,Nα/2G1,βNτ;α,N=1.74542-3.45537x99/100,N=10000,Nα/2G1,βNτ;α,N=1.74583-3.45689x99/100,N=100000.Notice that (45)P1,βτ;α=1.74588-3.45707x99/100,in accordance with (35).Furthermore,(46)G2,βτ;α,1000=4.9936·10-7x99/50-0.000469348x99/100+0.073817,G2,βτ;α,10000=1.68426·10-9x99/50-0.0000154412x99/100+0.0237096,G2,βτ;α,100000=5.64578·10-12x99/50-5.05717·10-7x99/100+0.00758763,G2x;1000=4.2320·10-7x2-0.000423206x+0.0704639,G2x;10000=1.34130·10-9x2-0.0000134131x+0.0223529,G2x;100000=4.24253·10-12x2-4.24253·10-7x+0.00707082,in accordance with (33).By using the above expressions for fractional Gram orthonormal polynomials, we obtain(47)Nα/2G2,βNτ;α,N=13.2866x99/50-13.3812x99/100+2.25505,N=1000,Nα/2G2,βNτ;α,N=13.3786x99/50-13.4487x99/100+2.26425,N=10000,Nα/2G2,βNτ;α,N=13.3882x99/50-13.4557x99/100+2.2652,N=100000.Notice that (48)P2,βτ;α=13.3894x99/50-13.4566x99/100+2.2653,in accordance with (35).


### 4.3. Interlacing Properties of the Zeros

Let us consider the polynomials obtained fromGn(x;α,β,N)=Gn,β(τ;α,N) after the change of variable x=τβ. 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 [0,N], and there exists at most one zero in each interval [i,i+1], i=0,…,N-1.As an example, for the specific values of the parametersα=3/5, β=1/7, and N= 100,000, we have obtained the following zeros of the above mentioned polynomials: (49)G2x;α,β,N=0,3.55906,4.95338,G3x;α,β,N=0,2.67741,4.15339,5.03948,G4x;α,β,N=0,1.93655,3.32757,4.43799,5.07999,G5x;α,β,N=0,0.978606,2.55054,3.71232,4.60351,5.10308.


## 4.1.α=3/5 and β=1/7

In this case,(36)G1,βτ;α,1000=-0.430804x7+1.05357,G1,βτ;α,10000=-0.158489x7+0.53835,G1,βτ;α,100000=-0.0573041x7+0.270416.Therefore, (37)Nα/2G1,βNτ;α,N=-9.18017x7+8.36879,N=1000,Nα/2G1,βNτ;α,N=-9.36325x7+8.53227,N=10000,Nα/2G1,βNτ;α,N=-9.3858x7+8.55131,N=100000.Notice that (38)P1,βτ;α=-9.39005x7+8.55459,in accordance with (35).Moreover,(39)G2,βτ;α,1000=0.753333x2/7-3.26517x7+3.42283,G2,βτ;α,10000=0.259692x2/7-1.58875x7+2.3636,G2,βτ;α,100000=0.0700715x2/7-0.596479x7+1.23532.As a consequence, (40)Nα/2G2,βNτ;α,N=43.0656x2/7-69.5788x7+27.1885,N=1000,Nα/2G2,βNτ;α,N=57.1894x2/7-93.8606x7+37.4605,N=10000,Nα/2G2,βNτ;α,N=59.4447x2/7-97.697x7+39.0641,N=100000.In this case, (41)P2,βτ;α=59.7007x2/7-98.1278x7+39.2423,also in accordance with (35).


## 4.2.α=99/100 and β=99/100

For these values ofα and β, we have that (42)G1,βτ;α,1000=0.0570034-0.000120685x99/100,G1,βτ;α,10000=0.0182768-3.9673·10-6x99/100,G1,βτ;α,100000=0.00584794-1.2992·10-7x99/100.Notice that (43)G1x;1000=0.0546902-0.00010938x,G1x;10000=0.0173179-3.46358·10-6x,G1x;100000=0.00547714-1.0954·10-7x,in accordance with (33).From the above expressions for fractional Gram orthonormal polynomials, we have(44)Nα/2G1,βNτ;α,N=1.74141-3.44076x99/100,N=1000,Nα/2G1,βNτ;α,N=1.74542-3.45537x99/100,N=10000,Nα/2G1,βNτ;α,N=1.74583-3.45689x99/100,N=100000.Notice that (45)P1,βτ;α=1.74588-3.45707x99/100,in accordance with (35).Furthermore,(46)G2,βτ;α,1000=4.9936·10-7x99/50-0.000469348x99/100+0.073817,G2,βτ;α,10000=1.68426·10-9x99/50-0.0000154412x99/100+0.0237096,G2,βτ;α,100000=5.64578·10-12x99/50-5.05717·10-7x99/100+0.00758763,G2x;1000=4.2320·10-7x2-0.000423206x+0.0704639,G2x;10000=1.34130·10-9x2-0.0000134131x+0.0223529,G2x;100000=4.24253·10-12x2-4.24253·10-7x+0.00707082,in accordance with (33).By using the above expressions for fractional Gram orthonormal polynomials, we obtain(47)Nα/2G2,βNτ;α,N=13.2866x99/50-13.3812x99/100+2.25505,N=1000,Nα/2G2,βNτ;α,N=13.3786x99/50-13.4487x99/100+2.26425,N=10000,Nα/2G2,βNτ;α,N=13.3882x99/50-13.4557x99/100+2.2652,N=100000.Notice that (48)P2,βτ;α=13.3894x99/50-13.4566x99/100+2.2653,in accordance with (35).


## 4.3. Interlacing Properties of the Zeros

Let us consider the polynomials obtained fromGn(x;α,β,N)=Gn,β(τ;α,N) after the change of variable x=τβ. 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 [0,N], and there exists at most one zero in each interval [i,i+1], i=0,…,N-1.As an example, for the specific values of the parametersα=3/5, β=1/7, and N= 100,000, we have obtained the following zeros of the above mentioned polynomials: (49)G2x;α,β,N=0,3.55906,4.95338,G3x;α,β,N=0,2.67741,4.15339,5.03948,G4x;α,β,N=0,1.93655,3.32757,4.43799,5.07999,G5x;α,β,N=0,0.978606,2.55054,3.71232,4.60351,5.10308.


## 5. Conclusions

By using fractional differences, we have introduced fractional Gram orthonormal polynomials. These polynomials converge asN→∞ to fractional Legendre orthonormal polynomials. Moreover, as α,β→1, they converge to “classical” Gram orthonormal polynomials. Finally, as N→∞ and α,β→1, 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 [i,i+1], i=0,1,…,N-1.



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*Source: 141325-2015-07-08.xml*