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ON THE LINK BETWEEN FINITE DIFFERENCE AND

DERIVATIVE OF POLYNOMIALS

KOLOSOV PETRO

Abstract.The main aim of this paper to establish the relations between for- ward, backward and central nite (divided) dierences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polyno- mials. Keywords.nite dierence, divided dierence, high order nite dierence, derivative, ode, pde, partial derivative, partial dierence, power, power func- tion, polynomial, monomial, power series, high order derivative

2010 Math. Subject Class.46G05, 30G25, 39-XX

e-mail:kolosov94@mail.ua

ORCID:http://orcid.org/0000-0002-6544-8880

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Contents

1. Introduction 2

2. Denitions forxgdistribution 3

3. Dierence and derivative of power function 4

4. Dierence of polynomials 6

5. Relation with Partial derivatives 8

6. Relations between nite dierences 10

7. The error of approximation 11

8. Summary 11

9. Conclusion 12

1

2 KOLOSOV PETRO

References 12

10. Appendix 1. Dierence table up to tenth power 13

1.Introduction

Let introduce the basic denition of nite dierence. Finite dierence is dif- ference between function values with constant increment. There are three types of nite dierences: forward, backward and central. Generally, the rst order forward dierence could be noted as:
hf(x) =f(x+h)f(x) backward, respec- tively, isrhf(x) =f(x)f(xh) and centralhf(x) =f(x+12 h)f(x12 h), whereh=const, (see [1], [2], [3]). When the increment is enough small, but constant, we can say that nite dierence divided by increment tends to deriv- ative, but not equals. The error of this approximation could be counted next:
hf(x)h f0(x) =O(h)!0, whereh- increment, such that,h!0. By means of induction as well right for backward dierence. More exact approximation we have using central dierence, that is: hf(x)h f0(x) =O(h2), note that function should be twice dierentiable. The nite dierence is the discrete analog of the derivative (see [4]), the main distinction is constant increment of the function's argument, while dierence to be taken. Backward and forward dierences are opposite each other. More generally, high order nite dierences (forward, backward and central, respectively) could be denoted as (see [7]): (1.1)
khf(x) =
k1f(x+h)
k1f(x) =nX k=0 n k (1)k
f(x+ (nk)h) (1.2)nhf(x) =nX k=0(1)kn k
f x+n2 k h (1.3)rkhf(x) =rk1f(x) rk1f(xh) =nX k=0 n k (1)k
f(xkh) Let describe the main properties of nite dierence operator, they are next (see [5]) (1)

Linearit yrules
( f(x) +g(x)) =
f(x) +
g(x)

(f(x) +g(x)) =f(x) +g(x) r(f(x) +g(x)) =rf(x) +rg(x) (2)
( C
f(x)) =C

f(x);r(C
f(x)) =C
rf(x); (C
f(x)) =C
f(x) (3)

Constan tr ule
C=rC=C= 0

Strictly speaking, divided dierence (see [6]) with constant increment is discrete analog of derivative, when nite dierence is discrete analog of function's dier- ential. They are close related to each other. To show this, let dene the divided dierence. Denition 1.4.Divided dierence of xed increment denition (forward, centaral, backward respectively) f +[xi; xj] :=f(xj)f(xi)x jxi; j > i;
x1 ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 3 f [xi; xj] :=f(xi)f(xj)x ixj; j < i;rx1 f c[xi] :=f(xi+m)f(xim)2m Hereby, divided dirence could be represented from the nite dierence, let be j=iconst, backward as, respectively + as forward andcas centered f [xi; xj]
f(xj)
xrf(xi)rx f c[xi]f(xim)2mf(xim)x

Then-order

f [xi; xj]n
nf(xj)
xnrnf(xi)rxn f c[xi]nnf(xim)(2m)nnf(xim)x n Each properties, which holds for nite dierences holds for divided dierences as well.

2.Definitions forxgdistribution

Let be variablexg:xg=g
C; C=xg+1xg= const; C2R>0!xg2 R >0; g2Z. To dene the nite dierence of function of such argument, we take C=hand rewrite forward, backward and central dierences of some analytically dened functionf(xi) next way:
f(xi+1) =f(xi+1)f(xi);rf(xi1) =f(xi) f(xi1); f(xi) =f x i+12  f x i12  . Then-th dierences of such a function could be written as (2.1)
nf(xi+1) =
n1f(xi+1)
n1f(xi) =nX k=0 n k (1)k
f(xi+nk) (2.2)nf(xi) =nX k=0 n k (1)k
fxi+n2 k
(2.3)rnf(xi1) =rn1f(xi) rn1f(xi1) =nX k=0 n k (1)k
f(xin+k) Let be dierences
f(xi+1); f(xi);rf(xi1), such thati2Zand dierences is taken starting from pointi, which divides the spaceZintoZ=Z[Z+symmetri- cally (note that +=symbols mean the left and right sides of start pointi= 0, i.e backward and forward direction), this way we have (i+1)2Z+;(i1)2Z; i6=

02(Z+;Z). Let derive some properties of that distribution:

(1) max( Z) = min(Z+) =i (2) F orwarddierence is tak enstarting from min( Z+), while backward from max(Z) (3) card( Z+) = card(Z), i.eP k2Z+1 =P k2Z1 (4) Maximal order of forw arddierence in w hichit is not equal to zero is max(Z+)

4 KOLOSOV PETRO

(5) Maximal order of bac kwarddierence in whic hit is not equal to zero is min(Z) (6) Maximal order of cen traldierence in whic hit is not equal to zero is max(Z+) (7) F orwardand bac kwarddierence equal eac hoth erb yabsolute v alue,while to be taken fromi= 0 Limitation 2.4.Note that most expression generated as case ofi= 0, so the initial start point of each dierence and inducted expressions are 0. Denitions 2.5.Generalized denitions complete this section (1)Z+:=N1- positive integers (2)Z:=f1;2; :::;min(Z)g- negative integers (3)ff; f(x); f(xi)g:=xn- power function, value of power function in point iof dierence table (4)i= 0 - initial point of every dierentiating process,f(x0) exist only for operator of centered dierence (as per limitation 2.4) (5)xi:=i

x rx(x)=2 =P
x- value of function's argument in point iof dierence table (6)
x rx(x)=2 - function's argument dierentials, constant values 2R>0 (7)
f(xi+1);rf(xi1) - forward and backward nite dierences in points i+ 1 andi1 of dierence table (8)f(xi) - centered nite dierence in pointiof dierence table (9)

0f0f r0ff

3.Difference and derivative of power function

Since then-order polynomial dened as summation of argument to power mul- tiplied by coecient, with higher powern, let describe a few properties of nite (divided) dierence of power function. Lemma 3.1.For each power function with natural number as exponent holds the equality between forward, backward and central divided dierences, and derivative with order respectively to exponent and equals to exponent under factorial sign mul- tiplied by argument dierential to power. Proof.Let be functionf(x) =xn; n2N. The derivative of power function,f0(x) = nx n1, sok-th derivativef(k)(x) =n
(n1)


(nk+ 1)
xnk,n > k. Using limit notation, we have: lim m!nf(m)(x) =f(n)(x) =n!:Let rewrite expressions (2.1, 2.2, 2.3) according to denitionxi=i

x, note that
x rxx=2. By means of power function multiplication property (i

x)n=in

xn, we can rewrite then-th nite dierence equations (2.1, 2.2, 2.3) as follows (3.2)
m(xni+1) =mX k=0 m k (1)k
i+mk
m

xm; m < n2N Using limit notation on divided by
xto power (3.2), we obtain (3.3) lim m!n
m(xni+1)
xm= lim m!nm X k=0 m k (1)k
i+mk
m ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 5 = nX k=0 n k (1)k
i+nk
n0=n! Similarly, going from (2.3), backwardn-th dierence equals: (3.4) lim m!nr m(xni1)rxm= lim m!nm X k=0 m k (1)k
im+k
m = nX k=0 n k (1)k
in+k
n0=n!

Andn-th central (2.2), respectively

(3.5) lim m!n mf(xi)x m= lim m!nm X k=0 m k (1)k
 i+m2 k m = nX k=0 n k (1)k
 i+n2 k n0=n!

As we can see the next conformities hold

(3.6) lim
x!0
nf
xnlim
x!C
nf
xnn! (3.7) lim x!0 nfx nlimx!C nfx nn! (3.8) lim rx!0r nfrxnlimrx!Cr nfrxnn! (3.9) lim
x!C
nf
xnlimx!C nfx nlimrx!Cr nfrxn8(C2R+)

In partial case whenC= 0

(3.10) dnfdx nlimx!0 nfx nlimrx!0r nfrxn

As well holds

(3.11) dfdx (x0) =limrx!0rfrx(x0) (3.12) dnfdx nlim
x!C
nf
xnlimx!C nfx nlimrx!Cr nfrxn;8(C2R+) wheref=xn. And there is exist the continuous derivative and dierence of order knsincef2Cnclass of smoothness. Thus, from (3.6, 3.7, 3.8), we can conclude (3.13) dnxndx n=
n(xni+1)
xn=n(xni)x n=rn(xni1)rxn=n!;(
x; x;rx)6!dx

This completes the proof.

6 KOLOSOV PETRO

Denition 3.14.We introduce the dierence equality operatorE(f), such that (3.15)E(f)def=
nf
xn=nfx n=rnfrxn Property 3.16.Let be central dierence written asmf(xi) =f(xi+m)f(xim) then-th central dierence ofn-th power isnm(xni) =n!
2m
xn, wherex= x i+1xi= const.

Going from lemma (3.1), we have next properties

(1)
k(xki+1) = const;(i+ 1)2Z+ : max(Z+)> k!
k(xki+1)
k(xki) (2)rk(xki1) = const;(i1)2Z:min(Z)k! rk(xki1) rk(xki) (3)k(xki) = const; i2Z+ : max(Z+)> k!k(xki)k(xki+j) (4)8([i+ 1]2Z+;[i1]2Z) :
k+j(xki+1) =rk+j(xki1) = 0; j >1;since
CC rC0 (5)8(f=xn; n2N; kn) :
kf= (1)n1
rkf. (6)
f(xi+1) =jrf(xi1)j (7)2f(x0) = 2
(x)n;8(f(xj) =xnj; nmod2 = 0) (8)8nmod2 = 0 :2j+1f(x0) = 0; j2N0(see Appendix 1 for reference) (9)8nmod2 = 1 :2jf(x0) = 0; j2N1 Hereby, according to above properties, we can write the lemma (3.1) for enough large setsZ+;Zas (3.17) dnxndx n=
n(xni)
xn=n(xni)x n=rn(xni)rxn=n! Or (3.18)ddx  n x n=E(xn) =n!

4.Difference of polynomials

Let be polynomialPn(xg) dened as

(4.1)Pn(xg) =nX i=0a ixig Finite dierences of such kind polynomial, are
Pn(xi) =Pn(xi+1)Pn(xi), rPn(xi1) =Pn(xi)Pn(xi1),Pn(xi) =Pn x i+12  Pn x i12  . Such way, according to the properties (1, 2, 3) from section 1, high order nite dierences of polynomials could be written as:
kPn(xi+1) =
k(a0
x0i+1+


+an
xni+1) =
k(a0
x0i+1) +


+
k(an
xni+1) (4.2) =a0

k(x0i+1) +


+an

k(xni+1)

Backward dierence, respectively, is

r kPn(xi1) =rk(a0
x0i1+


+an
xni1) =rk(a0
x0i1) +


+rk(an
xni1) (4.3) =a0
rk(x0i1) +


+an
rk(xni1)

And central

 kPn(xi) =k(a0
x0i+


+an
xni) =k(a0
x0i) +


+k(an
xni) ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 7 (4.4) =a0
k(x0i) +


+an
k(xni) Above expressions hold for each build naturaln-order polynomial. Lemma 4.5.8([i+ 1]2Z+;[i1]2Z) :
k+j(xki+1) rk+j(xki1)0; j1 Proof.According to lemma (3.1), then-th dierence ofn-th power is constant, consequently, the constant rule (3) holds
C=C=rC= 0. According to lemma (4.5) and properties (2, 3), taking the limits of (4.2, 4.3,

4.4), receive:

(4.6)
k!nPn(xi+1) = limk!n
k(a0
x0i+1) +


+
k(an
x0i+1) =
n(an
xni+1) =an

n(xni+1) (4.7)k!nPn(xi) = limk!nk(a0
x0i) +


+k(an
xni) =n(an
xni) =an
n(xni) (4.8)rk!nPn(xi1) = limk!nrk(a0
x0i1) +


+rk(an
xni1) =rn(an
xni1) =an
rn(xni1) Since then-th dierence ofn-th power equals ton!, we have theorem. Theorem 4.9.Eachn-order polynomial has the constantn-th nite (divided) dif- ference and derivative, which equals each other and equal constant timesn!, where nis natural. Proof.According to limits (4.6, 4.7, 4.8), we have
nPn(xi+1) =an

n(xni+1); r nPn(xi1) =an
rn(xni1); nPn(xi) =an
n(xni), going from lemma (3.1), the n-th dierence ofn-order polynomial equals tokn
n!, the properties (1, 2, 3, 4) proofs that for enough large setsZ+;Zwe have
n(xni+1)
n(xni); n(xni)  n(xni+j);rn(xni1) rn(xni);min(Z)nmax(Z+):Therefore, we have equality (4.10) dnPn(x)dx n=
nPn(xi)(
x)n=nPn(xi)(x)n=rnPn(xi)(rx)n=an
n!

Or, by means of denition (3.14) one has

(4.11) ddx  n P n(x) =E(Pn(x)) =an
E(xn)  Property 4.12.Let be a plot ofrkxni(k); i2Z(see Appendix 1, second line for reference)

8 KOLOSOV PETRO

02468100:500:51
1010k2[0;10]r

kx1010(k)Figure 1. Plot ofrkxni(k); i2Z It's seen that each k-order backward dierence (acc. to app 1) of powern, such that nkcould be well interpolated by means of general Harmonic oscillator equation (4.13)x=A0etsin(!t+'0)

Particularizing 4.13 we get

(4.14)rkxni(jk) =xneksin(!k+'0) In the points of local minimum and maximum ofRxneksin(!k+'0)dkwe have r kxn; k2[1;n]N1. By means of (5) we have relation with forward dierence (4.15)
kxni(k) = (1)n1xneksin(!k+'0)

Property 4.12 as well holds for polynomials.

5.Relation with Partial derivatives

Let be partial nite dierences dened as

(5.1)
f(u1; u2;:::; un)u1:=f(u1+h; u2;:::; un)f(u1; u2;:::; un) (5.2)f(u1; u2;:::; un)u1:=f(u1+h; u2;:::; un)f(u1h; u2;:::; un) (5.3)rf(u1; u2;:::; un)u1:=f(u1; u2;:::; un)f(u1h; u2;:::; un) By means of mathematical induction, going from Lemma (3.1), we have equal- ity betweenn-th partial derivative andn-th partial dierence, while be taken of polynomial dened function or power function. Theorem 5.4.For eachn-th natural power of many variables then-th partial divided dierences andn-th partial derivatives equal each other. Proof.Let be functionZ=f(u1; u2;:::; un) = (u1; u2;:::; un)n, where dots mean the general relations, i.e multiplication and summation between variables. ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 9 We denote the equality operator of partial dierence asE(F(u1; u2; :::; un))uk, whereukis variable of taken dierence. On this basis (5.5) @nZ@u nk=
nZuk
unk=nZuku nk=rnZukrunk=A
n!

Or, using equality operator

(5.6) @nZ@u nk=E(Z)uk=A
n! whereAis free constant, depending of relations between variables and 0k n. Property 5.7.Let be partial dierences of the functionf(u1;


;uk) =un1 u n2


unk; n2N;
f(u1;


;uk)M; f(u1;


;uk)M;rf(u1;


;uk)M, whereM- complete set of variables, i.eM=fuigkithen-th partial dierences of each variables are (5.8)
nf(u1; u2; u3; :::; uk)u1; u2; u3;:::;uk=k
n!
(
u1)n


(
uk)n (5.9)nf(u1; u2; u3; :::; uk)u1; u2; u3; :::;uk=k
n!
(u1)n


(uk)n (5.10)rnf(u1; u2; u3; :::; uk)u1; u2; u3; :::;uk=k
n!
(ru1)n


(ruk)n

8k2Z: max(Z+)> n >min(Z);(u1)(u2):::(uk);

(ru1)(ru2):::(ruk);(
u1)(
u2):::(
uk)

Otherwise

(5.11)
nf(u1; u2; u3; :::; uk)u1; u2; u3; :::;uk=n!
kX i=1(
ui)n (5.12)nf(u1; u2; u3; :::; uk)u1; u2; u3; :::;uk=n!
kX i=1(ui)n (5.13)rnf(u1; u2; u3; :::; uk)u1; u2; u3; :::;uk=n!
kX i=1(rui)n Note that here the partial dierences of non-single variable dened as
nf(u;

1:::;uk)M=
n1f(u1+h;:::;uk+h)M
n1f(u1;:::;uk)M

 nf(u1;:::;uk)M=n1f(u1+h;:::;uk+h)Mn1f(u1h;:::;ukh)M r nf(u1;:::;uk)M=rn1f(u1;:::;uk)M rn1f(u1h;:::;ukh)M Moreover, then-th partial dierence taken over enough large setZ+and8i:
xi= 1 has the next connection with single variablen-th derivative ofn-th power (5.14)
nf(u1; u2; u3; :::; uk)u1; u2; u3;:::;uk=kX i=1 ddu i n f(ui)

With partial derivative we have relation

(5.15)
nf(u1; u2; u3; :::; uk)u1; u2; u3;:::;uk=kX i=1 @@u i n f(u1; u2; u3; :::; uk)

10 KOLOSOV PETRO

Multiplied (5.14) and (5.15) by coecient, as dened, gives us relation withn-th partial polynomial. Theorem 5.16.For each non-single variable polynomial with ordernholds the equality betweenkn-order partial dierences and derivative.

Proof.Let be non-single variable polynomial

(5.17)Pn(un) =nX i=1M i
uii Going from property (5.7), thek-th partial dierences of one variable are (5.18)
kPn(un)uk=Mk
k!
(
uk)k; kPn(un)uk=Mk
k!
(uk)k; r kPn(un)uk=Mk
k!
(ruk)k

0kn. Thek-th partial derivative:

(5.19) @kPn(un)@u kk=Mk
k!

Hereby,

(5.20) @kPn(un)@u kk=
kPn(un)uk
ukk=kPn(un)uku kk=rkPn(un)ukrukk

Also could be denoted as

(5.21) @kPn(un)@u kk=E(Pn(un))uk=Mk
k!; kn

And completes the proof.

6.Relations between finite differences

In this section are shown relations between central, backward and central nite dierences, generally, they are (6.1)divf(x) :=f(x+
x)f(x
x)2

xdef=12  f(x+
x)
xf(x rx)rx =  f(x+
x) =
f(x) +f(x) f(x rx) =f(x) rf(x) =12 
f(x) +f(x)
xf(x) rf(x)rx = 12 
f(x) +rf(x)
x rx where "div" means divided, i.edivf(x) :=f(x)=(2

x). Hereby, (6.2) 2
divf(x)

x=
f(x) +rf(x)

And so on. Let be
x!0

(6.3) lim
x!02
divf(x)

x= 2
df(x) Or (6.4) 2
lim
x!0divf(x) = 2
df(x)dx !lim
x!0divf(x) =df(x)dx wheref(x) is power function, hence, the general relation between derivative and each kind nite dierence is reached, as desired. ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 11

7.The error of approximation

The error of derivative approximation done by forward nite dierence with respect to orderkncould be calculated as follows (7.1) 

x k x nddx  k x n=O(xnk)

Forn-order polynomial is

(7.2) 

x k P n(x)ddx  k P n(x) =O(xnk)

The partial, ifmk

(7.3)

uk m

Z@@u

k m

Z=O(ukm

k)

WhereO- Landau-Bachmann symbol (see [8], [9]).

8.Summary

In this section we summarize the obtained results in the previous chapters and establish the relationship between them. According to lemma (3.1), theorems (4.9), (5.4), (5.16) we have concluded (8.1) dnxndx n=E(xn) =n! (8.2) dnPn(x)dx n=E(Pn(x)) =an
E(xn) (8.3) @nZ@u nk=E(Z)uk=A
n! (8.4) @kPn(un)@u kk=E(Pn(un))uk=Mk
k! Generalizing these expressions, we can derive the general relations between ordinary, partial derivatives and nite (divided) dierences (8.5)E(un) =E(Pn(ug)) =E(Z)uk=E(Pn+j(un+j))un|{z} Y (8.6) dnundu n=dnPn(u)du n=@nZ@u nk=@nPn+j(un+j)@u nn|{z}

U; j0

8(A; Mn; an) = 1

I.e the equalities hold with precision to constant. FunctionZdened asZ= f(u1; u2;:::; un) = (u1; u2;:::; un)n. And nally Y=U with same limitations.

12 KOLOSOV PETRO

9.Conclusion

In this paper were established the equalities between ordinary and partial nite (divided) dierences and derivatives of power function and polynomials, with order equal between each other.

References

[1] P aulWilm ott;Sam Ho wison;Je Dewynne (1995). The Mathematics of Financial Deriv atives: A Student Introduction. Cambridge University Press. p. 137. ISBN 978-0-521-49789-3. [2] M. Hanif Chaudhry (2007). Op en-ChannelFlo w.Springer. p. 369. I SBN978-0-3 87-68648-6. [3] P eterOlv er(2013). In troductionto P artialDieren tialEquations . Springer Sc ienceand Busi- ness Media. p. 182. [4] W eisstein,Eric W. " FiniteDierence." F romMathW orld [5] D. Gleic h(2005), Finite Calculus: A T utorialfor Solving Nast ySums p 6-7. online cop y [6] Bakh valovN. S. Numerical Metho ds:Analysis, Algebra, Ordinary Dieren tialEquations p.

42, 1977.

[7] G. M. Fic htenholz(1968). Dieren tialand in tegralcalculus (V olume1). p. 244. [8] P aulBac hmann.Analytisc heZahlen theorie,v ol.2,Leipzig, T eubner189 4. [9] Edm undL andau.Handbuc hder Lehre v onder V erteilungder Primz ahlen,T eubner,Leipzig

1909, p.883.

ON THE LINK BETWEEN FINITE DIFFERENCE AND DERIVATIVE OF POLYNOMIALS 13

10.Appendix 1. Difference table up to tenth powerix

if(
;r)f=10f(
;r)2f=9f(
;r)3f=8f(
;r)4f=7f(
;r)5f=6f(
;r)6f=5f(
;r)7f=4f(
;r)8f=3f(
;r)9f=2f(
;r)10f=f-10-1010000000000-65132155994100173022-24783970201425878520-771309000385363440-17297280066528000-199584003628800

-9-93486784401-24130425771621776002-1052518500654569520-385945560212390640-10644480046569600-16329600-8926258176

-8-81073741824-791266575569257502-397948980268623960-173554920105945840-59875200302400007912982528-3204309152

-7-7282475249-222009073171308522-12932502095069040-6760908046070640-29635200-69591244802931599528-1013275648

-6-660466176-5070055141983502-3425598027459960-21538440164354406063636480-2668596480953858048-272709624

-5-59765625-87170497727522-67960205921520-5103000-52254720002415240960-895488000263003048-59417600 -4-41048576-989527931502-8745008185204443586560-2171473920838164480-25335552058370048-9706576 -3-359049-5802557002-55980-37158912001937295360-781885440243767040-573235209647528-1047552 -2-21024-102310223096576000-1703116800727695360-23417856056279040-95884801046528-59048 -1-11-1-18579456001703116800-619315200234178560-541900809588480-104448059048-1024

000371589120001238630400010838016002088960020480

1111185794560017031168006193152002341785605419008095884801044480590481024

2210241023102230965760001703116800727695360234178560562790409588480104652859048

3359049580255700255980371589120019372953607818854402437670405732352096475281047552

44104857698952793150287450081852044435865602171473920838164480253355520583700489706576

559765625871704977275226796020592152051030005225472000241524096089548800026300304859417600

666046617650700551419835023425598027459960215384401643544060636364802668596480953858048272709624

7728247524922200907317130852212932502095069040676090804607064029635200695912448029315995281013275648

881073741824791266575569257502397948980268623960173554920105945840598752003024000079129825283204309152

99348678440124130425771621776002105251850065456952038594556021239064010644480046569600163296008926258176

101010000000000651321559941001730222478397020142587852077130900038536344017297280066528000199584003628800

Note that central dierences divided byboldtypeset and kept in the middle of table. The table shows example in case maxZ+= 10;minZ=10;
x= 1; n= 10