On the link between finite differences and derivatives of polynomials

On the lin between finite differences and derivatives of polynomials Kolosov Petro To cite this version: Kolosov Petro. On the lin between finite differences and derivatives of polynomials. 13 pages, 1 figure. 2017. <hal-01350976v3> HAL Id: hal-01350976 https://hal.archives-ouvertes.fr/hal-01350976v3 Submitted on 8 Mar 2017 (v3), last revised 6 May 2017 (v4) HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives 4.0 International License

ON THE LINK BETWEEN FINITE DIFFERENCES AND DERIVATIVES OF POLYNOMIALS KOLOSOV PETRO Abstract. The main aim of this paper to establish the relations between forward, bacward and central finite(divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials. Keywords. finite difference, divided difference, high order finite difference, derivative, ode, pde, partial derivative, partial difference, power, power function, polynomial, power series, n-th difference of n-th power, high order derivative, discrete function 2010 Math. Subject Class. 46G05, 30G25, 39-XX e-mail: olosov 94@mail.ua Contents 1. Introduction 1 2. Definitions for x g distribution 3 3. Difference and derivative of power function 4 4. Difference of polynomials 6 5. Relation with Partial derivatives 8 6. Relations between finite differences 10 7. The error of approximation 10 8. Summary 11 9. Conclusion 11 References 11 10. Appendix 1. Difference table up to tenth power 12 1. Introduction Let introduce the basic definition of finite difference. Finite difference is difference between function values with constant increment. There are three types of finite differences: forward, bacward and central. Generally, the first order forward difference could be noted as: h f(x) = f(x+h) f(x) bacward, respectively, is h f(x) = f(x) f(x h) and central δ h f(x) = f(x+ 1 2 h) f(x 1 2 h), where h = const, (see [1], [2], [3]). When the increment is enough small, but constant, we can say that finite difference divided by increment tends to derivative, but not equals. The error of this approximation could be counted next: h f(x) h f (x) = O(h) 0, where h - increment, such that, h 0. By means of 1

2 KOLOSOV PETRO induction as well right for bacward difference. More exact approximation we have δ using central difference, that is: h f(x) h f (x) = O(h 2 ), note that function should be twice differentiable. The finite difference is the discrete analog of the derivative (see [4]), the main distinction is constant increment of the function s argument, while difference to be taen. Bacward and forward differences are opposite each other. More generally, high order finite differences (forward, bacward and central, respectively) could be denoted as (see [7]): (1.1) h f(x) = 1 f(x+h) 1 f(x) = ( 1) f(x+(n )h) n (1.2) δh n f(x) = ( ( n ) ) ( 1) f x+ 2 h (1.3) hf(x) = 1 f(x) 1 f(x h) = ( 1) f(x h) Let describe the main properties of finite difference operator, they are next (see [5]) (1) Linearity rules (f(x)+g(x)) = f(x)+ g(x) δ(f(x)+g(x)) = δf(x)+δg(x) (f(x)+g(x)) = f(x)+ g(x) (2) (C f(x)) = C f(x), (C f(x)) = C f(x), δ(c f(x)) = C δf(x) (3) Constant rule C = C = δc = 0 Strictly speaing, divided difference (see [6]) with constant increment is discrete analog of derivative, when finite difference is discrete analog of function s differential. They are close related to each other. To show this, let define the divided difference. Definition 1.4. Divided difference of fixed increment definition (forward, centaral, bacward respectively) f + [x i, x j ] := f(x j) f(x i ) x j x i, j > i, x 1 f [x i, x j ] := f(x i) f(x j ) x i x j, j < i, x 1 f c [x i ] := f(x i+m) f(x i m ) 2m Hereby, divided diffrence could be represented from the finite difference, let be j = i±const, bacward as, respectively + as forward and c as centered The n-order f ± [x i, x j ] f(x j) x f c [x i ] δf(x i±m) 2m f(x i) x δf(x i±m) δx f ± [x i, x j ] n n f(x j ) x n n f(x i ) x n

ON THE LINK BETWEEN FINITE DIFFERENCES AND DERIVATIVES OF POLYNOMIALS3 f c [x i ] n δn f(x i±m ) (2m) n δn f(x i±m ) δx n Each properties, which holds for finite differences holds for divided differences as well. 2. Definitions for x g distribution Let be variable x g : x g = g C, C = x g+1 x g = const, C R >0 x g R >0, g Z. To define the finite difference of function of such argument, we tae C = h and rewrite forward, bacward and central differences of some analytically defined function f(x i ) next way: f(x i+1 ) = f(x i+1 ) f(x i ), f(x i 1 ) = f(x i ) f(x i 1 ), δf(x i ) = f could be written as ( x i+ 1 2 ) f ( x i 1 2 (2.1) n f(x i+1 ) = n 1 f(x i+1 ) n 1 f(x i ) = (2.2) δ n f(x i ) = ). The n-th differences of such a function ( 1) f ( ) x i+ n 2 (2.3) n f(x i 1 ) = n 1 f(x i ) n 1 f(x i 1 ) = ( 1) f(x i+n ) ( 1) f(x i n+ ) Let be differences f(x i+1 ), δf(x i ), f(x i 1 ), such that i Z and differences is taen starting from point i, which divides the space Z into Z = Z Z + symmetrically (note that +/ symbols mean the left and right sides of start point i = 0, i.e bacward and forward direction), this way we have (i+1) Z +, (i 1) Z, i 0 (Z +,Z ). Let derive some properties of that distribution: (1) max(z ) = min(z + ) = i (2) Forward difference is taen starting from min(z + ), while bacward from max(z ) (3) card(z + ) = card(z ), i.e Z + 1 = Z 1 (4) Maximal order of forward difference in which it is not equal to zero is max(z + ) (5) Maximal order of bacward difference in which it is not equal to zero is min(z ) (6) Maximal order of central difference in which it is not equal to zero is max(z + ) (7) Forward and bacward difference equal each other by absolute value, while to be taen from i = 0 Limitation 2.4. Note that most expression generated as case of i = 0, so the initial start point of each difference and inducted expressions are 0. Definitions 2.5. Generalized definitions complete this section (1) Z + := N 1 - positive integers (2) Z := { 1, 2,..., min(z )} - negative integers (3) {f, f(x), f(x i )} := x n - power function, value of power function in point i of difference table

4 KOLOSOV PETRO (4) i = 0 - initial point of every differentiating process, δf(x 0 ) exist only for operator of centered difference (as per limitation 2.4) (5) x i := i x x (δx)/2 = x - value of function s argument in point i of difference table (6) x x (δx)/2 - function s argument differentials, constant values R >0 (7) f(x i+1 ), f(x i 1 ) - forward and bacward finite differences in points i+1 and i 1 of difference table (8) δf(x i ) - centered finite difference in point i of difference table (9) 0 f δ 0 f 0 f f 3. Difference and derivative of power function Since the n-order polynomial defined as summation of argument to power multiplied by coefficient, with higher power n, let describe a few properties of finite (divided) difference of power function. Lemma 3.1. For each power function with natural number as exponent holds the equality between forward, bacward and central divided differences, and derivative with order respectively to exponent and equals to exponent under factorial sign multiplied by argument differential to power. Proof. Letbefunctionf(x) = x n, n N. Thederivativeofpowerfunction, f (x) = nx n 1, so -th derivative f () (x) = n (n 1) (n +1) x n, n >. Using limit notation, we have: lim m n f (m) (x) = f (n) (x) = n!. Let rewrite expressions (2.1, 2.2, 2.3) according to definition x i = i x, note that x x δx/2. By means of power function multiplication property (i x) n = i n x n, we can rewrite the n-th finite difference equations (2.1, 2.2, 2.3) as follows (3.2) m (x n i+1 ) = m ( ) m ( 1) (i+m ) m x m, m < n N Using limit notation on divided by x to power (3.2), we obtain m (x n i+1 (3.3) lim ) m ( ) m m n x m = lim ( 1) (i+m ) m m n = ( 1) (i+n ) n 0 = n! Similarly, going from (2.3), bacward n-th difference equals: m (x n i 1 (3.4) lim ) m n x m = lim = And n-th central (2.2), respectively δ m f(x i ) (3.5) lim m n δx m = lim m m n ( ) m ( 1) (i m+ ) m ( 1) (i n+ ) n 0 = n! m m n ( ) m ( 1) (i+ m ) m 2

ON THE LINK BETWEEN FINITE DIFFERENCES AND DERIVATIVES OF POLYNOMIALS5 = As we can see the next conformities hold (3.6) lim x 0 ( ( 1) i+ n ) n 0 2 = n! n f x n lim x C n f x n n! δ n f (3.7) lim δx 0 δx n lim δ n f δx C δx n n! n f (3.8) lim x 0 x n lim n f x C x n n! n f (3.9) lim x C x n lim δ n f δx C δx n lim n f x C x n (C R + ) In partial case when C = 0 d n f (3.10) dx n lim δ n f δx 0 δx n lim n f x 0 x n As well holds df (3.11) dx (x 0) = lim f x 0 x (x 0) (3.12) d n f dx n lim n f x C x n lim δ n f δx C δx n lim n f x C x n, (C R+ ) where f = x n. And there is exist the continuous derivative and difference of order n since f C n class of smoothness. Thus, from (3.6, 3.7, 3.8), we can conclude d n x n (3.13) dx n = n (x n i+1 ) x n = δn (x n i ) δx n = n (x n i 1 ) x n = n!, ( x, δx, x) dx This completes the proof. Definition 3.14. We introduce the difference equality operator E(f), such that ( (3.15) E(f) def n ) f = x n = δn f δx n = n f x n Property 3.16. Let be central difference written as δ m f(x i ) = f(x i+m ) f(x i m ) the n-th central difference of n-th power is δ n m(x n i ) = n! 2m δxn, where δx = x i+1 x i = const. Going from lemma (3.1), we have next properties (1) (x i+1 ) = const, (i+1) Z+ : max(z+) > (x i+1 ) (x i ) (2) (x i 1 ) = const, (i 1) Z : min(z ) (x i 1 ) (x i ) (3) δ (x i ) = const, i Z+ : max(z+) > δ (x i ) δ (x i+j ) (4) ([i+1] Z+, [i 1] Z ) : +j (x i+1 ) = +j (x i 1 ) = 0, j > 1, since C δc C 0 (5) (f = x n, n N, n) : f = ( 1) n 1 f. (6) f(x i+1 ) = f(x i 1 ) (7) δ 2 f(x 0 ) = 2 (δx) n, (f(x j ) = x n j, n mod2 = 0) (8) n mod2 = 0 : δ 2j+1 f(x 0 ) = 0, j N 0 (see Appendix 1 for reference)

6 KOLOSOV PETRO (9) n mod2 = 1 : δ 2j f(x 0 ) = 0, j N 1 Hereby, according to above properties, we can write the lemma (3.1) for enough large sets Z +, Z as (3.17) Or (3.18) d n x n dx n = n (x n i ) x n = δn (x n i ) δx n = n (x n i ) x n = n! Let be polynomial P n (x g ) defined as d x n = E(x n ) = n! dx 4. Difference of polynomials (4.1) P n (x g ) = a i x i g Finite differences of such ind polynomial, are P n (x i ) = ( P n (x i+1 )) P n (x i ), P n (x i 1 ) = P n (x i ) P n (x i 1 ), δp n (x i ) = P n (x 1 i+ ) P n x 1 2 i. Such way, 2 according to the properties (1, 2, 3) from section 1, high order finite differences of polynomials could be written as: P n (x i+1 ) = (a 0 x 0 i+1 + +a n x n i+1 ) = (a 0 x 0 i+1 )+ + (a n x n i+1 ) (4.2) = a 0 (x 0 i+1)+ +a n (x n i+1) Bacward difference, respectively, is P n (x i 1 ) = (a 0 x 0 i 1 + +a n x n i 1) = (a 0 x 0 i 1)+ + (a n x n i 1) (4.3) = a 0 (x 0 i 1 )+ +a n (x n i 1 ) And central δ P n (x i ) = δ (a 0 x 0 i + +a n x n i ) = δ (a 0 x 0 i)+ +δ (a n x n i ) i=0 (4.4) = a 0 δ (x 0 i)+ +a n δ (x n i ) Above expressions hold for each build natural n-order polynomial. Lemma 4.5. ([i+1] Z +, [i 1] Z ) : +j (x i+1 ) +j (x i 1 ) 0, j 1 Proof. According to lemma (3.1), the n-th difference of n-th power is constant, consequently, the constant rule (3) holds C = δc = C = 0. According to lemma (4.5) and properties (2, 3), taing the limits of (4.2, 4.3, 4.4), receive: (4.6) n P n (x i+1 ) = lim n { (a 0 x 0 i+1 )+ + (a n x 0 i+1 )} = n (a n x n i+1) = a n n (x n i+1) (4.7) δ n P n (x i ) = lim n { δ (a 0 x 0 i )+ +δ (a n x n i )} = δ n (a n x n i ) = a n δ n (x n i )

ON THE LINK BETWEEN FINITE DIFFERENCES AND DERIVATIVES OF POLYNOMIALS7 (4.8) n P n (x i 1 ) = lim n { (a 0 x 0 i 1 )+ + (a n x n i 1 )} = n (a n x n i 1 ) = a n n (x n i 1 ) Since the n-th difference of n-th power equals to n!, we have theorem. Theorem 4.9. Each n-order polynomial has the constant n-th finite (divided) difference and derivative, which equals each other and equal constant times n!, where n is natural. Proof. According to limits (4.6, 4.7, 4.8), we have n P n (x i+1 ) = a n n (x n i+1 ), n P n (x i 1 ) = a n n (x n i 1 ), δn P n (x i ) = a n δ n (x n i ), going from lemma (3.1), the n-th difference of n-order polynomial equals to n n!, the properties (1, 2, 3, 4) proofs that for enough large sets Z +, Z we have n (x n i+1 ) n (x n i ), δn (x n i ) δ n (x n i+j ), n (x n i 1 ) n (x n i ), min(z ) n max(z + ). Therefore, we have equality (4.10) d n P n (x) dx n = n P n (x i ) ( x) n = δn P n (x i ) (δx) n = n P n (x i ) ( x) n = a n n! Or, by means of definition (3.14) one has d (4.11) P n (x) = E(P n (x)) = a n E(x n ) dx Property 4.12. Let be a plot of x n i (), i Z (see Appendix 1, second line for reference) 1 10 10 x 10 10() 0.5 0 0.5 0 2 4 6 8 10 [0;10] Figure 1. Plot of x n i (), i Z It s seen that each -order bacward difference (acc. to app 1) of power n, such that n could be well interpolated by means of general Harmonic oscillator equation (4.13) x = A 0 e βt sin(ωt+ϕ 0 )

8 KOLOSOV PETRO Particularizing 4.13 we get (4.14) x n i (j ) = x n e β sin(ω +ϕ 0 ) In the points of local minimum and maximum of x n e β sin(ω+ϕ 0 )d we have x n, [1; n] N 1. By means of (5) we have relation with forward difference (4.15) x n i () = ( 1)n 1 x n e β sin(ω +ϕ 0 ) Property 4.12 as well holds for polynomials. 5. Relation with Partial derivatives Let be partial finite differences defined as (5.1) f(u 1, u 2,..., u n ) u1 := f(u 1 +h, u 2,..., u n ) f(u 1, u 2,..., u n ) (5.2) δf(u 1, u 2,..., u n ) u1 := f(u 1 +h, u 2,..., u n ) f(u 1 h, u 2,..., u n ) (5.3) f(u 1, u 2,..., u n ) u1 := f(u 1, u 2,..., u n ) f(u 1 h, u 2,..., u n ) By means of mathematical induction, going from Lemma (3.1), we have equality between n-th partial derivative and n-th partial difference, while be taen of polynomial defined function or power function. Theorem 5.4. For each n-th natural power of many variables the n-th partial divided differences and n-th partial derivatives equal each other. Proof. Let be function Z = f(u 1, u 2,..., u n ) = (u 1, u 2,..., u n ) n, where dots mean the general relations, i.e multiplication and summation between variables. We denote the equality operator of partial difference as E(F(u 1, u 2,..., u n )) u, where u is variable of taen difference. On this basis (5.5) n Z u n Or, using equality operator = n Z u u n = δn Z u δu n = n Z u u n = A n! n Z (5.6) u n = E(Z) u = A n! where A is free constant, depending of relations between variables and 0 n. Property 5.7. Let be partial differences of the function f(u 1,,u ) = u n 1 ± u n 2 ± ±un, n N, f(u 1,,u ) M, δf(u 1,,u ) M, f(u 1,,u ) M, where M - complete set of variables, i.e M = {u i } i the n-th partial differences of each variables are (5.8) n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = ± n! ( u 1 ) n ( u ) n (5.9) δ n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = ± n! (δu 1 ) n (δu ) n (5.10) n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = ± n! ( u 1 ) n ( u ) n Z : max(z + ) > n > min(z ), (δu 1 ) (δu 2 )... (δu ), ( u 1 ) ( u 2 )... ( u ), ( u 1 ) ( u 2 )... ( u )

ON THE LINK BETWEEN FINITE DIFFERENCES AND DERIVATIVES OF POLYNOMIALS9 Otherwise (5.11) n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = n! (5.12) δ n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = n! (5.13) n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = n! ( u i ) n i=1 (δu i ) n i=1 ( u i ) n Note that here the partial differences of non-single variable defined as n f(u, 1...,u ) M = n 1 f(u 1 +h,...,u +h) M n 1 f(u 1,...,u ) M δ n f(u 1,...,u ) M = δ n 1 f(u 1 +h,...,u +h) M δ n 1 f(u 1 h,...,u h) M n f(u 1,...,u ) M = n 1 f(u 1,...,u ) M n 1 f(u 1 h,...,u h) M Moreover, the n-th partial difference taen over enough large set Z + and i : x i = 1 has the next connection with single variable n-th derivative of n-th power d (5.14) n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = f(u i ) du i With partial derivative we have relation (5.15) n f(u 1, u 2, u 3,..., u ) u1, u 2, u 3,...,u = i=1 u i i=1 i=1 f(u 1, u 2, u 3,..., u ) Multiplied (5.14) and (5.15) by coefficient, as defined, gives us relation with n-th partial polynomial. Theorem 5.16. For each non-single variable polynomial with order n holds the equality between n-order partial differences and derivative. Proof. Let be non-single variable polynomial (5.17) P n (u n ) = M i u i i Going from property (5.7), the -th partial differences of one variable are (5.18) P n (u n ) u = M! ( u ), δ P n (u n ) u = M! (δu ), 0 n. The -th partial derivative: (5.19) Hereby, (5.20) P n (u n ) u i=1 P n (u n ) u = M! ( u ) P n (u n ) u = P n (u n ) u u = M! = δ P n (u n ) u δu = P n (u n ) u u

10 KOLOSOV PETRO Also could be denoted as (5.21) P n (u n ) u And completes the proof. = E(P n (u n )) u = M!, n 6. Relations between finite differences In this section are shown relations between central, bacward and central finite differences, generally, they are (6.1) δ div f(x) := f(x+ x) f(x x) def = 1 ( f(x+ x) f(x x) ) 2 x 2 x x f(x+ x) = f(x)+f(x) = f(x x) = f(x) f(x) = 1 ( f(x)+f(x) f(x) f(x) ) 2 x x = 1 2 ( ) f(x)+ f(x) x x where div means divided, i.e δ div f(x) := δf(x)/(2 x). Hereby, (6.2) 2 δ div f(x) x = f(x)+ f(x) And so on. Let be x 0 (6.3) lim x 0 2 δ divf(x) x = 2 df(x) Or (6.4) 2 lim δ divf(x) = 2 df(x) lim x 0 dx δ divf(x) = df(x) x 0 dx where f(x) is power function, hence, the general relation between derivative and each ind finite difference is reached, as desired. 7. The error of approximation The error of derivative approximation done by forward finite difference with respect to order n could be calculated as follows ( ) ( ) d (7.1) x n x n = O(x n ) x dx For n-order polynomial is ( ) (7.2) P n (x) x ( ) d P n (x) = O(x n ) dx The partial, if m ( ) m ( ) m (7.3) Z Z = O(u m ) u u Where O - Landau-Bachmann symbol (see [8], [9]).

ON THE LINK BETWEEN FINITE DIFFERENCES AND DERIVATIVES OF POLYNOMIALS11 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) (8.2) (8.3) d n x n dx n = E(xn ) = n! d n P n (x) dx n = E(P n (x)) = a n E(x n ) n Z u n = E(Z) u = A n! P n (u n ) (8.4) u = E(P n (u n )) u = M! Generalizing these expressions, we can derive the general relations between ordinary, partial derivatives and finite (divided) differences (8.5) E(u n ) = E(P n (u g )) = E(Z) u = E(P n+j (u n+j )) un }{{} Y (8.6) d n u n du n = dn P n (u) du n = n Z u n = n P n+j (u n+j ) u n, j 0 }{{} U (A, M n, a n ) = 1 I.e the equalities hold with precision to constant. Function Z defined as Z = f(u 1, u 2,..., u n ) = (u 1, u 2,..., u n ) n. And finally with same limitations. Y = U 9. Conclusion In this paper were established the equalities between ordinary and partial finite (divided) differences and derivatives of power function and polynomials, with order equal between each other. References [1] Paul Wilmott; Sam Howison; Jeff Dewynne (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press. p. 137. ISBN 978-0-521-49789-3. [2] M. Hanif Chaudhry (2007). Open-Channel Flow. Springer. p. 369. ISBN 978-0-387-68648-6. [3] Peter Olver (2013). Introduction to Partial Differential Equations. Springer Science and Business Media. p. 182. [4] Weisstein, Eric W. Finite Difference. From MathWorld [5] D. Gleich (2005), Finite Calculus: A Tutorial for Solving Nasty Sums p 6-7. online copy [6] Bahvalov N. S. Numerical Methods: Analysis, Algebra, Ordinary Differential Equations p. 42, 1977. [7] G. M. Fichtenholz (1968). Differential and integral calculus (Volume 1). p. 244. [8] Paul Bachmann. Analytische Zahlentheorie, vol.2, Leipzig, Teubner 1894. [9] Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909, p.883.

12 KOLOSOV PETRO 10. Appendix 1. Difference table up to tenth power i x i f (, )f/δ 10 f (, ) 2 f/δ 9 f (, ) 3 f/δ 8 f (, ) 4 f/δ 7 f (, ) 5 f/δ 6 f (, -10-10 10000000000-6513215599 4100173022-2478397020 1425878520-771309000 385-9 -9 3486784401-2413042577 1621776002-1052518500 654569520-385945560 212-8 -8 1073741824-791266575 569257502-397948980 268623960-173554920 105-7 -7 282475249-222009073 171308522-129325020 95069040-67609080 46-6 -6 60466176-50700551 41983502-34255980 27459960-21538440 16-5 -5 9765625-8717049 7727522-6796020 5921520-5103000 -522-4 -4 1048576-989527 931502-874500 818520 4443586560-217 -3-3 59049-58025 57002-55980 -3715891200 1937295360-78 -2-2 1024-1023 1022 3096576000-1703116800 727695360-23 -1-1 1-1 -1857945600 1703116800-619315200 234178560-54 0 0 0 3715891200 0 1238630400 0 108380160 1 1 1 1 1857945600 1703116800 619315200 234178560 54 2 2 1024 1023 1022 3096576000 1703116800 727695360 234 3 3 59049 58025 57002 55980 3715891200 1937295360 781 4 4 1048576 989527 931502 874500 818520 4443586560 217 5 5 9765625 8717049 7727522 6796020 5921520 5103000 522 6 6 60466176 50700551 41983502 34255980 27459960 21538440 16 7 7 282475249 222009073 171308522 129325020 95069040 67609080 46 8 8 1073741824 791266575 569257502 397948980 268623960 173554920 105 9 9 3486784401 2413042577 1621776002 1052518500 654569520 385945560 212 10 10 10000000000 6513215599 4100173022 2478397020 1425878520 771309000 385 Note that central differences divided by bold typeset and ept in the middle of table. The table shows