# Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series

Authors: Vasily E. Tarasov
Journal: Journal of Mathematics
 (2015)

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## Abstract

New differences of integer orders, which are connected with derivatives of integer orders not approximately, are proposed. These differences are represented by infinite series. A characteristic property of the suggested differences is that its Fourier series transforms have a power-law form. We demonstrate that the proposed differences of integer ordersn are directly connected with the derivatives ∂n/∂xn. In contrast to the usual finite differences of integer orders, the suggested differences give the usual derivatives without approximation.


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## Body

## 1. Introduction

It is well known that the finite difference of integer ordern cannot be considered as an exact discretization of the derivative of order n. For example, the forward finite difference Δh1 of first order with step h can be formally represented [1, 2] by the Taylor series:(1)Δh1=h∂∂x+h22∂2∂x2+h36∂3∂x3+⋯=exp⁡h∂∂x-I.Therefore the finite differences of order n∈N give only approximation of the derivatives of integer orders ∂n/∂xn, and Δhn/hn gives the derivative at h→0 only. Using formal inverting of (1), we get the equation(2)h∂∂x=log⁡1+Δh1=∑j=0∞-1jj+1Δhj+1,which holds in the sense that left and right sides of (2) give the same result when applied to polynomials and analytic functions. Equation (2) allows us to assume that the derivatives of integer orders should be represented by differences that are represented by infinite series with power-law coefficients. For this reason, these differences will be called the infinite but they are not supposed infinite in value.It should be noted that infinite differences and corresponding derivatives of noninteger order have been first proposed by Grünwald [3] in 1867 and independently by Letnikov [4] in 1868. These differences are defined by infinite series (see Section  20 in [5]) as a generalization of the usual finite difference of integer orders. Now these differences are called the Grünwald-Letnikov fractional differences [5, 6].In this paper, we suggest new differences of integer orders that are connected with derivatives not approximately. A characteristic property of the suggested differences is that its Fourier series transforms have the power-law form. We demonstrate that these differences of integer ordersn are directly connected with the derivatives ∂n/∂xn. The main advantage of these differences is a direct connection with the usual derivatives for all integer orders.


## 2. Derivatives and Differences of Integer Orders

In this section we briefly describe the finite differences and derivatives of integer orders to fix notation for further consideration.


### 2.1. Finite Differences and Derivatives

Let us consider well-known finite differences of integer ordersn∈N.Definition 1.
The forward finite differenceΔhn of order n∈N with the step h is given by(3)Δhnfx≔∑m=0n-1mn!m!n-m!fx+n-mh.The backward finite difference ∇hn of order n∈N is given by(4)∇hnfx≔∑m=0n-1mn!m!n-m!fx-mh.The central finite difference δhn of order m∈N is given by(5)δhnfx≔∑m=0n-1mn!m!n-m!fx+n2-mh.These finite differences of order m∈N for functions f[n] of a discrete variable n∈Z are defined by the equations(6)Δmfn≔∑j=0m-1jm!j!m-j!fn+m-j,(7)∇mfn≔∑j=0m-1jm!j!m-j!fn-j,(8)δmfn≔∑j=0m-1jm!j!m-j!fn+m2-j.Let us give a connection of the finite differences and derivatives of integer orders.Proposition 2.
The finite differences (3)–(5) of order n∈N with the step h can be represented by the equations(9)Δhnfx=∑m=0n-1mn!m!n-m!exp⁡n-mh∂∂xfx,∇hnfx=∑m=0n-1mn!m!n-m!exp⁡-mh∂∂xfx,δhnfx=∑m=0n-1mn!m!n-m!exp⁡n2-mh∂∂xfx.Proof.
Using the well-known relation [7] of the form(10)exp⁡h∂∂xfx=fx+h,we get(11)fx+n-mh=exp⁡n-mh∂∂xfx,fx-mh=exp⁡-mh∂∂xfx,fx+n2-mh=exp⁡n2-mh∂∂xfx.Substituting (11) into the finite differences (3)–(5) give representation (9).Remark 3.
For example, the central finite difference of second order is represented [7] in the form(12)δh2=2∑j=1∞h2j12j!∂2j∂x2j.It is easy to see that(13)1h2δh2fx≠∂2fx∂x2. Only the limit h→0 gives(14)limh→0⁡δh2fxh2=∂2fx∂x2. As a result, we have that the central finite difference δh2 can be considered only as an approximation of the derivative of second order. Similarly, we have the inequality(15)1hnδhnfx≠∂nfx∂xnn∈N,and the equality for the limit h→0:(16)limh→0⁡δhnfxhn=∂nfx∂xn.For other finite differences (3)-(4) we have equations, which are similar to (15) and (16).Using representations (9) and (15), we can see that all finite differences (3)–(5) of orders n∈N cannot give exactly the derivative ∂n/∂xn. These differences can be considered only as approximation of the derivative of integer order n which coincide with derivatives at h→0 only.


### 2.2. Fourier Series Transform and Finite Differences

A conclusion about connection between finite differences and derivatives can also be obtained by using the Fourier series and integral transforms of differences (6)–(8) and derivatives.The Fourier integral transform of the functionf(x) is(17)f~k=∫-∞+∞dxfxe-ikx=Ffx,and the inverse Fourier integral transform is defined by(18)fx=12π∫-∞+∞dkf~keikx=F-1f~k.It is well known that the derivatives of integer ordern∈N have the Fourier integral transform in the form(19)F∂nfx∂xn=∫-∞+∞dxe-ikx∂nfx∂xn=iknf~k.The Fourier series transform of the discrete functionf[n]∈l2 is defined by(20)f^k=∑n=-∞+∞fne-iknh=Fh,Δfn.And the inverse Fourier series transform is(21)fn=h2π∫-π/h+π/hdkf^keiknh=Fh,Δ-1f^k.Note that we use the minus sign in the exponent of (20) instead of plus that is usually used.The Fourier series transforms of the finite differences of ordern∈N are given by the following proposition.Proposition 4.
The Fourier series transforms of differences (6)–(8) have the form(22)Fh,ΔΔmfn=∑j=0m-1jm!j!m-j!exp⁡ikm-jhf^k,(23)Fh,Δ∇mfn=∑j=0m-1jm!j!m-j!exp⁡-ikjhf^k,(24)Fh,Δδmfn=∑j=0m-1jm!j!m-j!exp⁡ikm2-jhf^k.Proof.
Let us consider the forward finite difference (6). Using (20), we get(25)Fh,ΔΔmfn=∑j=0m-1jm!j!m-j!∑n=-∞+∞e-iknhfn+m-j=∑j=0m-1jm!j!m-j!exp⁡ikm-jh∑n′=-∞+∞e-ikn′hfn′=∑j=0m-1jm!j!m-j!exp⁡ikm-jhf^k.As a result, we proved (22). Similarly, we get (23) and (24).Remark 5.
It is easy to see that the Fourier series transform of differences (6)–(8) of order n∈N cannot be considered as n power of (ihk):(26)Fh,ΔΔn≠ikhn.Only the limit h→0 gives(27)limh→0⁡Fh,ΔΔnhn=ikn.For other finite differences (4)-(5), we have equations, which are similar to (26) and (27).Proposition 6.
The inverse Fourier integral transform (18) of expressions (22)–(24) has the form(28)F-1Fh,ΔΔmfn=∑j=0m-1jm!j!m-j!exp⁡m-jh∂∂xfx,(29)F-1Fh,Δ∇mfn=∑j=0m-1jm!j!m-j!exp⁡-jh∂∂xfx,(30)F-1Fh,Δδmfn=∑j=0m-1jm!j!m-j!exp⁡m2-jh∂∂xf^k.Proof.
Let us consider (22) for the forward difference (6). Using the inverse Fourier integral transform (18) for the function expik(m-j)h, we get(31)F-1exp⁡ikm-jh=12π∫-∞+∞dkexp⁡ikm-jheikx=12π∑s=0∞m-jhss!∫-∞+∞dkikseikx=∑s=0∞m-jhss!∂s∂xs=exp⁡m-jh∂∂x.As a result, we proved (28). Similarly, we get (29) and (30).Remark 7.
It is easy to see that expressions (22)–(24) coincide with (11) by the change of variables j→m and m→n.Remark 8.
For example, the central finite difference (8) of second order has the Fourier series transform in the form(32)Fh,Δδ2=2∑j=1∞-1j2j!kh2j. Using (19), the inverse Fourier integral transform F-1 gives(33)F-1Fh,Δδ2=2∑j=1∞12j!h2j∂2j∂x2j,which coincides with (12).
As a result, we have that the characteristic property of finite difference (6) is the inequality(34)Fh,ΔΔn≠ikhn. This inequality leads us to the corresponding inequality(35)1hnF-1Fh,ΔΔn≠∂n∂xn, which means that finite difference (6) of orders n∈N cannot give exactly the derivative ∂n/∂xn. Only in the limit h→0, we get(36)limh→0⁡F-1Fh,ΔΔnhn=∂n∂xn. Therefore difference (6) can be considered only as approximation of the derivative ∂n/∂xn. For the finite differences ∇n and δn, which are defined by (7) and (8), we have the same equations.


## 2.1. Finite Differences and Derivatives

Let us consider well-known finite differences of integer ordersn∈N.Definition 1.
The forward finite differenceΔhn of order n∈N with the step h is given by(3)Δhnfx≔∑m=0n-1mn!m!n-m!fx+n-mh.The backward finite difference ∇hn of order n∈N is given by(4)∇hnfx≔∑m=0n-1mn!m!n-m!fx-mh.The central finite difference δhn of order m∈N is given by(5)δhnfx≔∑m=0n-1mn!m!n-m!fx+n2-mh.These finite differences of order m∈N for functions f[n] of a discrete variable n∈Z are defined by the equations(6)Δmfn≔∑j=0m-1jm!j!m-j!fn+m-j,(7)∇mfn≔∑j=0m-1jm!j!m-j!fn-j,(8)δmfn≔∑j=0m-1jm!j!m-j!fn+m2-j.Let us give a connection of the finite differences and derivatives of integer orders.Proposition 2.
The finite differences (3)–(5) of order n∈N with the step h can be represented by the equations(9)Δhnfx=∑m=0n-1mn!m!n-m!exp⁡n-mh∂∂xfx,∇hnfx=∑m=0n-1mn!m!n-m!exp⁡-mh∂∂xfx,δhnfx=∑m=0n-1mn!m!n-m!exp⁡n2-mh∂∂xfx.Proof.
Using the well-known relation [7] of the form(10)exp⁡h∂∂xfx=fx+h,we get(11)fx+n-mh=exp⁡n-mh∂∂xfx,fx-mh=exp⁡-mh∂∂xfx,fx+n2-mh=exp⁡n2-mh∂∂xfx.Substituting (11) into the finite differences (3)–(5) give representation (9).Remark 3.
For example, the central finite difference of second order is represented [7] in the form(12)δh2=2∑j=1∞h2j12j!∂2j∂x2j.It is easy to see that(13)1h2δh2fx≠∂2fx∂x2. Only the limit h→0 gives(14)limh→0⁡δh2fxh2=∂2fx∂x2. As a result, we have that the central finite difference δh2 can be considered only as an approximation of the derivative of second order. Similarly, we have the inequality(15)1hnδhnfx≠∂nfx∂xnn∈N,and the equality for the limit h→0:(16)limh→0⁡δhnfxhn=∂nfx∂xn.For other finite differences (3)-(4) we have equations, which are similar to (15) and (16).Using representations (9) and (15), we can see that all finite differences (3)–(5) of orders n∈N cannot give exactly the derivative ∂n/∂xn. These differences can be considered only as approximation of the derivative of integer order n which coincide with derivatives at h→0 only.


## 2.2. Fourier Series Transform and Finite Differences

A conclusion about connection between finite differences and derivatives can also be obtained by using the Fourier series and integral transforms of differences (6)–(8) and derivatives.The Fourier integral transform of the functionf(x) is(17)f~k=∫-∞+∞dxfxe-ikx=Ffx,and the inverse Fourier integral transform is defined by(18)fx=12π∫-∞+∞dkf~keikx=F-1f~k.It is well known that the derivatives of integer ordern∈N have the Fourier integral transform in the form(19)F∂nfx∂xn=∫-∞+∞dxe-ikx∂nfx∂xn=iknf~k.The Fourier series transform of the discrete functionf[n]∈l2 is defined by(20)f^k=∑n=-∞+∞fne-iknh=Fh,Δfn.And the inverse Fourier series transform is(21)fn=h2π∫-π/h+π/hdkf^keiknh=Fh,Δ-1f^k.Note that we use the minus sign in the exponent of (20) instead of plus that is usually used.The Fourier series transforms of the finite differences of ordern∈N are given by the following proposition.Proposition 4.
The Fourier series transforms of differences (6)–(8) have the form(22)Fh,ΔΔmfn=∑j=0m-1jm!j!m-j!exp⁡ikm-jhf^k,(23)Fh,Δ∇mfn=∑j=0m-1jm!j!m-j!exp⁡-ikjhf^k,(24)Fh,Δδmfn=∑j=0m-1jm!j!m-j!exp⁡ikm2-jhf^k.Proof.
Let us consider the forward finite difference (6). Using (20), we get(25)Fh,ΔΔmfn=∑j=0m-1jm!j!m-j!∑n=-∞+∞e-iknhfn+m-j=∑j=0m-1jm!j!m-j!exp⁡ikm-jh∑n′=-∞+∞e-ikn′hfn′=∑j=0m-1jm!j!m-j!exp⁡ikm-jhf^k.As a result, we proved (22). Similarly, we get (23) and (24).Remark 5.
It is easy to see that the Fourier series transform of differences (6)–(8) of order n∈N cannot be considered as n power of (ihk):(26)Fh,ΔΔn≠ikhn.Only the limit h→0 gives(27)limh→0⁡Fh,ΔΔnhn=ikn.For other finite differences (4)-(5), we have equations, which are similar to (26) and (27).Proposition 6.
The inverse Fourier integral transform (18) of expressions (22)–(24) has the form(28)F-1Fh,ΔΔmfn=∑j=0m-1jm!j!m-j!exp⁡m-jh∂∂xfx,(29)F-1Fh,Δ∇mfn=∑j=0m-1jm!j!m-j!exp⁡-jh∂∂xfx,(30)F-1Fh,Δδmfn=∑j=0m-1jm!j!m-j!exp⁡m2-jh∂∂xf^k.Proof.
Let us consider (22) for the forward difference (6). Using the inverse Fourier integral transform (18) for the function expik(m-j)h, we get(31)F-1exp⁡ikm-jh=12π∫-∞+∞dkexp⁡ikm-jheikx=12π∑s=0∞m-jhss!∫-∞+∞dkikseikx=∑s=0∞m-jhss!∂s∂xs=exp⁡m-jh∂∂x.As a result, we proved (28). Similarly, we get (29) and (30).Remark 7.
It is easy to see that expressions (22)–(24) coincide with (11) by the change of variables j→m and m→n.Remark 8.
For example, the central finite difference (8) of second order has the Fourier series transform in the form(32)Fh,Δδ2=2∑j=1∞-1j2j!kh2j. Using (19), the inverse Fourier integral transform F-1 gives(33)F-1Fh,Δδ2=2∑j=1∞12j!h2j∂2j∂x2j,which coincides with (12).
As a result, we have that the characteristic property of finite difference (6) is the inequality(34)Fh,ΔΔn≠ikhn. This inequality leads us to the corresponding inequality(35)1hnF-1Fh,ΔΔn≠∂n∂xn, which means that finite difference (6) of orders n∈N cannot give exactly the derivative ∂n/∂xn. Only in the limit h→0, we get(36)limh→0⁡F-1Fh,ΔΔnhn=∂n∂xn. Therefore difference (6) can be considered only as approximation of the derivative ∂n/∂xn. For the finite differences ∇n and δn, which are defined by (7) and (8), we have the same equations.


## 3. Formulation of the Problem

Finite differences of ordersn∈N can be considered only as approximation of the derivative of integer order n. We would like to get a difference that can be connected with the derivative ∂n/∂xn not only by the limit h→0.In order that a difference of ordern∈N, which will be denoted by ΔnT, corresponds to the derivative ∂n/∂xn not approximately, this difference should satisfy the condition(37)1hnF-1Fh,ΔΔnT=∂n∂xn.This condition can be realized if this difference has the Fourier series transform in the form(38)Fh,ΔΔnT=ikhn.Using (38), we can get an exact expression of the required difference ΔnT.Let us consider a form of differencesΔnT in order n∈N to obtain an exact expression of these differences by using condition (38). In the notation of the difference, T denotes that this difference gives the fractional derivative by transform operator T=F-1∘Fh,Δ by (37) (see also [8–12]).At the beginning, we will consider differencesΔn,±T with even and odd coefficients for simplification.Definition 9.
A general form of differencesΔn,±T of order n∈N is defined by the equation(39)Δn,±Tfm≔∑j=-∞+∞Kn±jfm-j,where the functions f[m]∈lq (q≥2) and the kernels Kn±(m)∈lp (p≥1) are the real-valued functions of discrete variable m∈Z such that the properties(40)Kn±-m=±Kn±mhold for all m∈N and n∈N. The operators Δn,±T will be called even (“+”) and odd (“−”) T-differences of order n.Remark 10.
In the definition we use the sequence spacelp (p>0), which is the linear space consisting of all discrete functions (sequences) f[n], where n∈Z, satisfying the inequality(41)∑n=-∞+∞fnp<∞. If p≥1, then we can define a norm on the lp-space by the equation(42)fp≔∑n=-∞+∞fnp1/p. The sequence space lp with p>0 is a complete metric space with respect to this norm, and therefore it is a Banach space. We can assume that f[n] belongs to the Hilbert space l2 of square-summable sequences to apply the Fourier series transform. It is known that if 1≤p<q, then lq⊂lp. Therefore, lq⊂l2 if q>2, and we will consider f[m]∈lq with q≥2.Proposition 11.
Difference (39), which are defined by convolutions of Kn±(m)∈lp(p>1) and f[m]∈lq(q≥2) are operators lq→lr that map the discrete function f[m]∈lq(q≥2) into functions g±[m]∈lr(r≥2) such that(43)g±m≔Δn,±Tfm∈lr, where m∈Z, and(44)1r+1=1p+1q.Proof.
It is known that iff[m]∈lq and K[m]∈lp, then the inequality(45)K∗fr≤Kpfq holds, where r is defined by (44) and the star ∗ denotes the convolution. This is Young’s inequality for convolutions (see [13, 14] and Theorem  276 of [15]). Using Young’s inequality in the form(46)Δn,±Tfmr=Kn±∗fr≤Kn±pfq, we get(47)g±m≔Δn,±Tfm∈lr if condition (44) holds and f[m]∈lq (q≥2) and Kn±(m)∈lp (p>1).Remark 12.
The Fourier series transforms of difference (39) have the form(48)Fh,ΔΔn,±Tfm≔∑m=-∞+∞e-ikmΔn,±Tfm=K^n±khf^k.In order to get (38), we should use(49)K^n+k=kn,K^n-k=isgn⁡kkn.Using property (40), the kernels Kn±(m) can be defined by the equations(50)Kn+m=1π∫0πK^n+kcos⁡kmdk,Kn-m=iπ∫0πK^n-ksin⁡kmdk.These equations allow us to get exact expressions for the kernels Kn±(m).


## 4. Exact Expression of InfiniteT-Differences

The exact expressions for the kernelsKm±(n), which satisfy conditions (49), are given by the following proposition.Proposition 13.
The kernelsKm±(n) of even and odd T-difference (39) for order m∈N, which satisfy conditions (49), are given by the equations(51)Km+n=∑k=0m-1/2-1n+km!πm-2k-2m-2k-1!1n2k+2+-1m+1/2m!2m+1/2-mπnm+1n∈N,(52)Km-n=-∑k=0m/2-1n+k+1m!πm-2k-1m-2k!1n2k+1--1m/2m!2m/2-m+1πnm+1n∈N,(53)Km±-n=±Km±nn∈N,Km+0=πmm+1,Km-0=0.Proof.
Substitution of (49) into (50) gives(54)Kn+m=1π∫0πkncos⁡kmdk,Kn-m=-1π∫0πknsin⁡kmdk.Then we use equation 2.5.3.5 of [16] that has the form(55)∫0πxmcos⁡nxdx=-1n+2nm+1∑k=0m-1/2-1km!m-2k-1!πnm-2k-1+-1m+1/2m!nm+12m+12-m,n∈N,∫0πxmsin⁡nxdx=-1n+1nm+1∑k=0m/2-1km!m-2k!πnm-2k+-1m/2m!nm+12m2-m+1,n∈N,where [x] is the integer part of the value x and n∈N. Here 2[(m+1)/2]-m=1 for odd m and 2[(m+1)/2]-m=0 for even m. As a result, we get (51), (52), and (53).Examples. The kernels (51) with m=2 and m=4 have the form(56)K2+n=2-1nn2n≠0,n∈Z,K2+0=π23,K4+n=4π2-1nn2-24-1nn4n≠0,n∈Z,K4+0=π45. Using (52) for m=1 and m=3, we get the examples(57)K1-n=-1nnn≠0,n∈Z,K1-0=0,K3-n=-1nπ2n-6-1nn3n≠0,n∈Z,K3-0=0.Let us defineT-difference ΔnT of order n∈N, that is, a generalization of Δn,±T such that conditions (37) and (38) hold.Definition 14.
A general form ofT-difference ΔnT of order n∈N is defined by the equation(58)ΔnTfm≔∑j=-∞+∞Knjfm-j,where the function f[m]∈lq (q≥2) and the kernel Kn(m)∈lp (p≥1) are real-valued function of integer variable m∈Z, and(59)Knm=cos⁡πn2Kn+m+sin⁡πn2Kn-m.Remark 15.
Substitution of (54) into (59) gives(60)Knm=1π∫0πkncos⁡πn2cos⁡km-sin⁡πn2sin⁡kmdk. As a result, we get(61)Knm=1π∫0πkncos⁡km+πn2dk.Remark 16.
Using (59), we can represent the T-difference through the even and odd differences in the form(62)ΔnTfm=cos⁡πn2Δn,+Tfm+sin⁡πn2Δn,-Tfm.Examples. Let us give exact equations for T-differences of orders n=1,2,3,4. The T-difference of first order has the form(63)Δ1Tfn≔∑m=-∞m≠0+∞-1mmfn-m.The T-difference of second order has the form(64)Δ2Tfn≔-∑m=-∞m≠0+∞2-1mm2fn-m-π23fn.The T-difference of third order(65)Δ3Tfn≔-∑m=-∞m≠0+∞-1mπ2m-6-1mm3fn-m.The T-difference of fourth order(66)Δ4Tfn≔∑m=-∞m≠0+∞4π2-1mm2-24-1mm4fn-m+π45fn.The minus is used in (64), (66) and the plus is used in (63), (66), since we take into the values cos⁡(πn/2) and sin(πn/2) for the order n∈{1,2,3,4}.Remark 17.
Note that the numerical series of (63)–(66) converge for f[m]∈lq (q≥2) since the kernels Kn(m)∈lp (p>1) (see Proposition 11). Note that using equation 5.1.2.3 of [16], we can get(67)∑m=1∞-1mms=21-s-1ζs=-1Γs∫0∞xs-1ex+1dx=Ts, where ζ(s) is the Riemann zeta function, Γ(s) is the Gamma function, and(68)T1=-ln⁡2,T2=-π212,T3=-0,90154268…,T4=-7π4720. As a result, series (63)–(66) with f[m]=1 for all m∈Z converge.Remark 18.
Let us note that we can consider theT-difference for n=-1. In this case, we have(69)ΔnT≔sin⁡-π2Δ-1,-T=-Δ-1,-T, where(70)K-1m=-K-1-m=+1π∫0πk-1sin⁡mkdk=1πSiπm, and Si(z) is the sine integral. As a result, we have the difference of first negative order in the form(71)Δ-1Tfn≔∑m=-∞m≠0+∞π-1Siπmfn-m.The Fourier series transform Fh,Δ of this difference is(72)Fh,ΔΔ-1T=ikh-1. This difference corresponds to the integration. Note that a possibility to use the infinite differences for negative orders is not unique. For example, the Grünwald-Letnikov derivatives can be used for orders α<0 (see Section  20 in [5] and Section  2.2 in [6]) if the functions f(x) satisfy the condition(73)fx<c1+x-μ,where μ>|α|.


## 5. Properties of InfiniteT-Differences

Let us give examples, which allows us to demonstrate that theT-difference Δ1T action is similar to derivative, when applied to polynomials and analytic functions.Proposition 19.
TheT-differences Δ1T of first order of the power-law function, the sine and cosine functions, and the exponential function have the form(74)Δ1Tnm=mnm-1m∈N,(75)Δ1Tsin⁡kn=kcos⁡knk∈R,(76)Δ1Tcos⁡kn=-ksin⁡knk∈R,(77)Δ1Texp⁡kn=kexp⁡knk∈R,where n∈N.Proof.
The proof is realized by direct calculation of series (63), where we use the Cesaro summations and Poisson-Abel summation (see Section  449 of [17]) for (74) in the form(78)∑m=1+∞-1m=-12,∑m=1+∞-1mm2j=0j∈N and the usual formulas for the product of trigonometric functions and equation 5.4.2.6 of [16] in the form(79)∑m=1∞-1mmsin⁡km=-k2.The connection ofT-difference ΔnT with the even and odd differences Δn,±T is given by the following proposition.Proposition 20.
TheT-difference ΔnT of order n∈N gives the even and odd T-differences Δn,±T by the equations(80)Δ2mT=-1mΔ2m,+T,Δ2m+1T=-1mΔ2m+1,-T,where m∈N.Proof.
For evenn=2m, where m∈N, the T-difference is given by(81)Δ2mT=cos⁡πmΔ2m,+T+sin⁡πmΔ2m,-T=-1mΔ2m,+T. For odd n=2m+1, where m∈N, the T-difference is(82)Δ2m+1T=cos⁡πm+π2Δ2m+1,+T+sin⁡πm+π2Δ2m+1,-Tfm=cos⁡πmΔ2m+1,-T=-1mΔ2m+1,-T.As a result, we proved (80).TheT-difference of order n∈N has the following Fourier transform.Proposition 21.
The Fourier series transformFh,Δ of the T-difference has the form(83)Fh,ΔΔnTfmk=khnf^k=ikhnf^k,where sgn⁡(k) is the sign of the variable k.Proof .
Using (62) and Euler’s formula, we get (84)Fh,ΔΔnTfmk=cos⁡πn2Fh,ΔΔn,+Tfmk+sin⁡πn2Fh,ΔΔnTfmk=cos⁡πn2K^n+kh+isgn⁡ksin⁡πn2K^n-khnFh,Δfk=cos⁡πn2khn+isgn⁡ksin⁡πn2khnFh,Δfk=eiπnsgn⁡kh/2khnf^k=eiπsgn⁡kh/2nkhnf^k=isgn⁡khnkhnf^k=ikhnf^k.Remark 22.
The main property of theT-differences is the Fourier series transform in the form(85)Fh,ΔΔnTfm=ikhnf~k.As a result, we can consider the inverse Fourier integral transform F-1 of (85) to get(86)F-1Fh,ΔΔnT=F-1ikhn=hn∂n∂xn.Let us prove that theT-differences ΔnT of integer orders n∈N are directly related with the derivatives ∂n/∂xn of orders n.Proposition 23.
The transform operatorsF-1∘Fh,Δ map the T-differences ΔnT of integer orders n∈N into the derivative of orders n:(87)F-1∘Fh,ΔΔnT=hn∂n∂xn.Proof.
Using the Fourier series transform (83) of T-difference, we get(88)F-1∘Fh,ΔΔnT=F-1ikhn=hn∂n∂xn.As a result, we have that the characteristic property of suggestedT-differences is the equality(89)Fh,ΔΔnT=ikhn that leads us to the corresponding equality(90)1hnF-1Fh,ΔΔnT=∂n∂xn, which means that the T-difference of order n∈N gives the derivative ∂n/∂xn exactly. The T-difference of order n∈N is connected with the derivative ∂n/∂xn not only asymptotically by the limit h→0. In the limit h→0, we also get(91)limh→0⁡F-1Fh,ΔΔnThn=∂n∂xn. Therefore the suggested T-difference can be considered not only as approximation of the derivative ∂n/∂xn, but it can also be considered as an exact discrete analog of derivatives of integer orders.


## 6. Conclusion

In this paper, we propose new type of differences that are directly connected with derivatives of integer order in contrast with usual finite differences that can be considered only as approximation of these derivatives. A main characteristic of the suggested differences is that its Fourier series transforms have the power-law form. In the paper we demonstrate that these differences of integer ordersn∈N are directly connected with the derivatives ∂n/∂xn. This is the main advantage of these differences in contrast with all other differences that can be considered as an approximation of integer order derivatives. It allows us to consider an exact discretization of differential equations. We assume that the suggested differences can be interesting in application since they allow us to reflect characteristic properties of complex continua at the microstructural and nanostructural levels by matching lattice models with long-range interactions.



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*Source: 134842-2015-11-04.xml*