1 Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace

We study discrete functions on equidistant and non-equidistant infinitesimal grids. We consider its difference quotients of higher order and give conditions for their near-equality to the corresponding derivatives. Important tools are the formula of Faa di Bruno for higher order derivatives and a discrete version of it. As an application of such transitions from the discrete to the continuous we extend the DeMoivre-Laplace Theorem to higher order: n-th order difference quotients of the binomial probability distribution tend to the corresponding n-th order partial differential quotients of the Gaussian distribution.


Introduction
Let δξ > 0 be infinitesimal in the sense of nonstandard analysis.We let X be the set of all multiples of δξ by some integer.We study discrete "quasi-continuous" functions f on this set of equally spaced points, but also on more irregular grids, which are images of X by some near-standard function, say φ.Such grids will be called near-continua.
Obviously, the well-known nonstandard notions of S-continuity and S-differentiability can be defined for near-continua, and we will extend these notions to regular behavior of all standard orders.This will lead to the notion of functions "of class S n ".This notion deals with the regularity of higher-order difference quotients much as the notion of functions of class C n deals with the regularity (here also existence) of higher-order derivatives.
We show that on equidistant near-continua X the difference quotients δ k f δξ k of functions of class S n are for k ≤ n nearly equal to the derivatives d k • f dx k of the shadow • f of For functions defined on non-equidistant near-continua φ(X) such nearequality does not need to hold -even for functions as elementary as quadratic functions -and appears to depend on the nature of the function φ.As is to be expected the Chain Rule will have some importance, in particular its version for higher order derivatives, known as the Formula of Faà di Bruno.We establish an approximate, discrete version of this formula.With the help of this formula we show that the difference quotients of standard order n of a function ψ : φ(X) → R are infinitely close to the corresponding derivatives of its shadow, provided that φ and ψ are both of class S n .
The present article extends the results on transitions between the discrete and the continuous on equidistant near-continua of [4], [11] and [6] to transitions on general near-continua.These studies ( [11] representing a Masters Thesis supervised by the author) apply such results to the transition of the discrete binomial probability distribution to the continuous Gaussian distribution, extending the DeMoivre-Laplace theorem in a sense to difference quotients and (partial) derivatives of higher order.We end the present article by a short proof of this extension.
The article has the following structure.Nonstandard analysis disposes of a terminology common to both a class of discrete functions and a class of continuous functions; this terminology facilitates the transition between discreteness and continuity and will be recalled in Section 2. In Section 3 we define the notion of a function of class S n for general near-continua.We show how these functions behave for discrete differentiation and integration.Also, the class S n is stable for algebraic operations on equidistant near-continua, but we provide an elementary counterexample for non-equidistant nearcontinua.Later on, at the end of Section 5, conditions will be given for algebraic operations to hold on (suitable parts of) such near-continua.In Section 4 we recall the tools for the transition from discreteness to continuity (the class of standard functions of class C n ) at arbitrary order on equidistant near-continua.In Section 5 we develop the tools for the transition from discreteness to continuity at arbitrary order on general near-continua.It is here that we derive a nonstandard, discrete and approximative version of the formula of Faà di Bruno.In Section 6 we extend the process of continuisation to functions of two variables.
Imme van den Berg Definition 2.1 Let A ⊂ R and f : A → R. Let x ∈ A. The function f is said to be S-continuous at x if for all y ∈ A x y ⇒ f (x) f (y).
The function f is said to be S-continuous on A if f is S-continuous at all x ∈ A.
is limited and S-continuous at all limited x ∈ A.
A class of obvious examples of functions of class S 0 are the standard everywhere continuous functions.
By no means do functions of class S 0 need to be limited for unlimited x: standard unbounded functions such as standard polynomials, as well as the exponential function, are all of class S 0 .Standard rational functions with poles, like f (x) = 1 1+x are of class S 0 on all sets A which do not contain elements infinitely close to the poles.
The next simple proposition often permits to verify that nonstandard functions are of class S 0 .Proposition 2.3 Let A ⊂ R and f , g : A → R. If f is of class S 0 and f (x) g(x) for all limited x ∈ A, then g is also of class S 0 .Proof Clearly g(x) is limited for all limited x ∈ A. Let x ∈ A be limited and y ∈ A, y x.Then g(y) f (y) f (x) g(x).
Hence g is S-continuous at x.
In this article we consider for most of the time functions defined on a discrete subset of R, consisting of successive points at an infinitesimal distance.

Definition 2.4
We let δξ always be a positive non-zero infinitesimal and is the image of an equidistant near-continuum by a strictly monotone function φ : X → R of class S 0 .
For convenience we suppose, unless otherwise said, that φ is increasing, unbounded from below and unbounded from above.
Definition 2.5 Let Y =φ(X) be a near-continuum, where φ : X → R is a strictly increasing function of class S 0 .Let a, b ∈ Y with a < b.We write Let η ∈ Y and let ξ ∈ X be such that η = φ(ξ).Sometimes we write η ξ = φ(ξ) and δη ξ = δφ(ξ), and we may even write with abuse of notation δη instead of δη ξ ; note that generally speaking δη is not a constant.Given a subset A ⊆ Y we define Notice that if f is a real function defined on A ⊆ X, the n th -order difference quotient or n th -order discrete derivative of f We give two examples of functions of class S 0 defined on an equidistant near-continuum X ≡ {kδξ |k ∈ Z }.The nonstandard function E : X → R defined by is limited and S-continuous for all limited x.Indeed, one proves easily that the Euler formula E(ξ) e ξ holds for all limited ξ ∈ X, and then one may apply Proposition 2.3.
A second nonstandard example is the function Then it is easy to verify that F(ξ) e ξ 2 /2 for all limited ξ ∈ X + .Again it follows from Proposition 2.3 that F is of class S 0 on X + .The discrete functions E and F have even more regularity, as will be shown in Section 3.
The class of functions of class S 0 is closed under the usual algebraic operations, provided one does not divide by infinitesimal values.The proof of the next proposition is straightforward.

Imme van den Berg
Proposition 2.6 Let A ⊂ R and f , g : A→ R be functions of class S 0 .Then −f , f + g and f • g are of class S 0 , and f /g is of class S 0 on all sets B ⊂ A such that g 0 on the limited part of B.
To functions of class S 0 , defined on a possibly discrete subset of R, may be associated standard continuous functions.This transition from the discrete to the continuous, or continuisation, uses the notion of shadow.
Any limited number y is nearly-equal to a unique standard number x called the standard part or the shadow of y, and we write x = • y.For instance, if 0, one has • = 0 and • e = 1.To each function f of class S 0 defined on R one may associate a unique standard function • f such that f (x) • f (x) for all limited x.The function • f is called the standard part or shadow of f .In addition, the function • f is continuous.This property was already known to Robinson [16].We state a version of this theorem for discrete functions on a near-continuum.
Theorem 2.7 (Theorem of the continuous shadow, one variable) Let Y be a nearcontinuum.Let f : Y → R be of class S 0 .Then there exists a unique standard function • f : R → R such that f (y) • f (y) for all limited y ∈ Y.In fact f is everywhere continuous.
It is this theorem which yields a general procedure for the transformation of discrete functions into continuous functions.For a proof of the theorem we refer to the literature [16][14] [8].

Functions of class S n of one variable
We extend the notion of functions of class S 0 to functions of class S n for all standard n ∈ N. Functions of class S n are not only limited and S-continuous themselves, but also their difference quotients of order m for m ≤ n.We show that discrete derivation and integration relate functions of class S n and S n+1 in a similar way as ordinary derivation and integration relate functions of class C n and C n+1 .Stability of the (external) set of functions of class S n appears to hold for all the usual algebraic operations on equidistant near-continua.On general near-continua stability may depend on the nature of the near-continuum.Hence on equidistant near-continua polynomials of standard degree and limited coefficients are of class S n , and rational functions which are quotients of such polynomials are of class S n whenever the arguments are not infinitely close to a singularity of the denominator, while such properties may not hold on all near-continua.
Journal of Logic & Analysis 5:6 (2013) Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace 7 Notation 3.1 We let Y =φ(X) be a near-continuum, where φ : X → R is a strictly increasing function of class S 0 .Let ψ : Y → R. Let η ∈ Y and let ξ ∈ X be such that η = φ(ξ).Depending on the context we use the following notations for the differences of ψ : We write Again depending on the context we define Observe that, if ψ is defined on A ⊆ Y and A (n) = ∅, its n th -order difference quotient ψ [n] is defined on A (n) .
A real function is of class C n if it is n times continuously differentiable.We define functions of class S n by analogy.
Definition 3.2 Let A⊆ Y and ψ : A→ R, and let n ∈ N, n ≥ 1 be standard.Assume A (n) = ∅.We say that ψ is of class S n on A if ψ [k] : The next proposition is a first consequence of Definition 3.2.
Proposition 3.3 Let A⊂ Y and let n ∈ N, n ≥ 1 be standard.Assume A (n) = ∅.Let ψ : A→ R be of class S n on A and assume 0 ≤ m < n.Then ψ is of class S m on A.
We now show that the difference quotient ψ [1] of a function ψ of class S n is of class S n−1 , implying that the difference quotient ψ [m] of order m < n is a function of class S n−m .Conversely, if the difference quotient of a function ψ of class S 0 is of class S n−1 , the function ψ will be of class S n .
For convenience we consider functions defined on the whole of Y.
Lemma 3.4 (Lemma of the discrete derivative) Let n ∈ N be standard and ψ : Y → R be of class S n .Then ψ [1] is of class S n−1 .
Imme van den Berg Proposition 3.5 Let m, n ∈ N be standard with m ≤ n and let ψ : Next lemma bears some relation to the following well-known theorem relating the convergence of sequences of functions and their derivatives: Given a sequence of real functions (f n ) n∈N , if the sequence of derivatives (f n ) n∈N converges uniformly to a real function g, and if at some point x 0 the sequence (f n (x 0 )) n∈N converges, the functions (f n ) n∈N converge uniformly to a function f with f = g.
Lemma 3.6 (Lemma of the discrete integral) Let n ∈ N be standard and ψ : Y → R.
If ψ [1] is of class S n and ψ(y 0 ) is limited for some limited y 0 ∈ Y, then ψ is of class S n+1 .
As a consquence we obtain a sort of minimal condition for functions to be of class S n .
We now consider the stability of the class S n under algebraic operations.The proof of the first proposition is immediate.We formulate the product rule and the division rule first for functions defined on an equidistant near-continuum X.Then we discuss its validity on general near-continua.Note that Proposition 3.9 Let n ∈ N be standard and f , g : Proof By external induction.The case n = 0 follows from Proposition 2.6.Assume the property is valid for some standard integer n.Suppose f and g are of class S n+1 Lemma 3.10 Let n ∈ N be standard and f : X → R be a function of class S n with appreciable values for limited arguments.Let A ⊆ X be such that Proof By external induction.The case n = 0 follows from Proposition 2.6.Assume the property is valid for some standard integer n.Let f be a function of class S n+1 and A ⊆ X be such that A (n+1) = ∅ and f takes appreciable values for limited arguments ξ ∈ A. Now [1] (ξ)δξ .
By Proposition 3.3 the function f is of class S n on A and by the Lemma of the discrete derivative f [1] is of class S n on A (1) .Then f 2 (ξ) − f (ξ)f [1] (ξ)δξ is of class S n on A (1) by Proposition 3.9.Then 1/(f 2 (ξ) − f (ξ)f [1] (ξ)δξ) is of class S n on A (1) by the Imme van den Berg induction hypothesis.Hence δ(1/f ) δξ is of class S n on A (1) .Because 1 f is of class S 0 on A, the function 1 f is of class S n+1 on A as a consequence of the Lemma of the discrete integral.
Proposition 3.11 Let n ∈ N be standard and f , g : X → R be functions of class S n .Let A ⊆ X be such that A (n) = ∅ and g 0 on the limited part of A.Then f /g is of class S n on A.
The following example shows somewhat surprisingly that a very elementar function does not need to be of class S 2 on too irregular grids.
Example 3.12 shows also that the product rule does not hold in all generality: if f and g are of class S n on some set Y for some limited n, the product f • g does not need to be of class S n .It is easy to verify that the division rule also does not hold in general.Indeed, with f (x) = 1, g(x) = x, χ(x) ≡ f (x)/g(x) = 1/x and φ as above one finds χ [2] (1 Hence χ [2] (1) is unlimited again for δξ = √ δξ .We will show in Subsection 5.2 that the product rule and the division rule do hold provided Y is the image of X by some function φ which is of class S n itself.
On the other hand it follows from the Propositions 3.8 and 3.9 that polynomials of standard degree with limited coefficients defined on the equidistant continuum X are of class S n for all standard n.Then by Proposition 3.11 a rational function which is a quotient of polynomials of standard degree and limited coefficients is of class S n on all sets of the form A (n) = ∅ such that A ⊆ X and no element of A is infinitely close to a limited singularity of the denominator.
We verify that the two examples of discrete functions E and F defined in Section 2 are of class S n for all standard n.
and E is of class S 0 , its discrete derivative δE δξ is also of class S 0 .This implies that E is of class S 1 .Then δE δξ is also class S 1 , so by the Lemma of the discrete integral E is of class S 2 .With external induction, applying this procedure to the induction step, one shows that the function E is of class S n for all standard n ∈ N.

Secondly, for ξ
We already saw that F is of class S 0 on X + .Because the monomial ξ is of class S n for all standard n and these classes are stable under multiplication, we may apply the same method as above to show that F is of class S n on X + , for all standard n.
4 Transition from the discrete to the continuous in one variable on equidistant near-continua Let X be an equidistant near-continuum as above.We consider first discrete functions of class S n on the whole of X and extend the Theorem of the continuous shadow to transitions from the discrete to the continuous of higher order of regularity.
The case for differentiability of first order had already been proved earlier [8] [4].
Then its shadow f is a real function of class C 1 and φ [1] (ξ) f (ξ) for all limited ξ ∈ X. [8] is different of ours.The essential difference is that it concerns only functions defined on R.However, this difference is not very important since a discrete function defined on X can always be appropriately extended to a real function f of class S 1 defined on the whole of R.

N.B. The definition of function of class S 1 in
The proof that the shadow of a function of class S n defined on X is a function of class C n is contained in [11] and [6] and will be repeated here.
Theorem 4.2 Let n ∈ N be standard and let φ : X → R be of class S n .Then its shadow is a function f : R → R of class C n and φ [n] (ξ) f (n) (ξ) for all limited ξ ∈ X.
Proof By external induction.If n = 0, by the Theorem of the continuous shadow f is a real function of class C 0 and φ(ξ) f (ξ) for all limited ξ ∈ X. Assume the property is valid for some standard integer n.Let φ be a function of class S n+1 .By Proposition 3.5 the function φ [n] is of class S 1 .By the induction hypothesis its shadow equals f (n) .By the Theorem of the differentiable shadow f (n) is continuously differentiable and φ As a consequence we obtain that for pairs of functions of class S n the property of near-equality is hereditary to discrete derivatives up to order n.Theorem 4.3 Let n ∈ N be standard and f , g : X → R be two functions of class S n such that f (ξ) g(ξ) for all limited ξ ∈ X.Then f [n] (ξ) g [n] (ξ) for all limited ξ ∈ X.
Proof By the Theorem of the continuous shadow Journal of Logic & Analysis 5:6 (2013) Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace 13 5 Transition from the discrete to the continuous on general near-continua We consider now functions of class S n on possibly non-equidistant near-continua.The next theorem gives conditions for near-equality of such functions and their difference quotients to their shadows and the respective derivatives of these shadows.
Theorem 5.1 Let n ∈ N be standard.Let ξ ∈ X be limited.Let φ : X → R be a function of class S n such that φ [1] (ξ) 0. Let ψ : φ(X) → R be a function of class S n .Let χ = ψ • φ.Then there exists a near-interval [α • •β] with α, β ∈ X limited and α ξ β such that φ [1] (ξ) 0 for all ξ ∈ [α • •β], and the function χ is of class (3) Formula (3) is known as the formula of Faà di Bruno.It extends the Chain Rule to derivatives of higher order.Thus formula (2) is a discrete, approximative version of the formula of Faà di Bruno.
One of the consequences of the theorem above is that a sufficient condition for a function ψ to be of class S n on some near-continuum Y ⊂ R, with the n th -order difference quotient infinitely close to the n th -order derivative of its shadow, is that Y is the image of an equidistant near-continuum X ⊂ R by a (locally) strictly monotone function φ which is itself of class S n .We already saw that absence of such a regularity condition may even lead to difference quotients with infinitely large values, which certainly are not nearly-equal to derivatives of • ψ .A near-continuum Y ⊂ R which is the image of an equidistant near-continuum X ⊂ R by a monotone function φ of class S n will be called a near-continuum of class S n .
Below we prove formula (2) along the lines of the proof of De la Vallée-Poussin (see also [12]) of the usual formula of Faà di Bruno (3).

Imme van den Berg
The proof of De la Vallée-Poussin is by induction.The first step consists in proving that the coefficients of g (k 1 +k 2 +...+kn) (f (x)) n i=1 are integers, and independent of f and g.This means that the coefficients may be determined for convenient special functions, in fact powers of polynomials of the form (a 1 x + a 2 x 2 + ... + a n x n ) k .This step uses the multinomial expansion Finally the powers of polynomials are repeatedly differentiated.
In the discrete case some complications arise from the fact that difference quotients are not always taken at ξ , but also at points ξ + θδξ , with θ > 0. Hence the product of powers at the end of the formula of Faà di Bruno transforms into a product of products; but we will show that the latter products are infinitely close to the corresponding powers taken at ξ .

Properties of the composition function
Convention, notations.Let n ∈ N be standard.Let ξ ∈ X be limited.Let φ : X → R be a function of class S n such that φ [1] (ξ) 0. As a consequence of the principle of Fehrele [8] there exists a near-interval [α • •β] with α, β ∈ X limited and α ξ β such that φ [1] (ξ) Without restriction of generality we assume that φ [1] We use the following notations.Let ψ : φ(X) → R be a function of class S n and let χ = ψ • φ.All functions being of class S 0 , their shadows are well-defined and we put The discrete Chain Rule for the first-order difference quotient of the composite function χ takes the form (4) χ [1] (ξ) = ψ [1] (φ(ξ))φ [1] (ξ).
This formula is used in the proof that the composition of two functions of class S n is also of class S n . .By the Lemma of the discrete derivative φ [1] is of class S n−1 on [α • •β] and ψ [1] is of class . By the Induction Hypothesis ψ [1] • φ is of class S n−1 on [α • •β] and then χ [1] is of class S n−1 on [α • •β] by ( 4) and by Proposition 3.9.Then it follows from the Lemma of the discrete integral that χ is of class We already saw that not on all near-continua φ(X) the class S n is stable under the usual algebraic operations.We show that this stability does hold for functions defined on near-continua of class S n .
We continue to assume the conventions which where introduced above and prove first a lemma.Proof The case n = 1 follows from the fact that δη ξ is infinitesimal: the number η ξ + δη ξ is limited whenever η ξ is limited.Assume that n > 1.By the Lemma of the discrete derivative f [1] is of class S n−1 in η and η [1] is of class S n−1 in ξ .Also g(η ξ ) = f η ξ + δη ξ = f [1] η ξ η [1] (ξ)δξ + f η ξ .
Then by Proposition 5.2 and Proposition 3.9 the function χ defined by χ(ξ) = f [1] η ξ η [1] (ξ) is of class which is of class S 0 in η .Then g [k] is of class S 0 in η by Proposition 2.3.Hence g is of class S n in η .Proof By external induction.The case n = 0 is contained in Proposition 2.6.Assume the property is valid for some standard integer n.Suppose f , g and η are of class S n+1 .
Hence (f • g) [1] is of class S n .Because f • g is of class S 0 , the Lemma of discrete integration implies that the function f • g is of class S n+1 .
Next lemma is proved similarly, now using the formula The proposition implies that, for all standard n, all polynomials of standard degree with limited coefficients defined on some near-continuum Y of class S n are of class S n , and all rational functions which are quotients of such polynomials are of class S n on all sets of the form A (n) = ∅, such that the elements of A ⊆ Y are not infinitely close to a limited singularity of the denominator.
As already said such functions do not need to be of class S n for all standard n ∈ N on too irregular near-continua φ(X).Next theorem says that a sufficiently sparse subset S of a near-continuum φ(X) of class S 0 may be determined which happens to be a near-continuum of class S n for all standard n ∈ N. Then the restrictions of the functions mentioned beforehand to S will be of class S n for all standard n ∈ N indeed.
Theorem 5.7 Let φ(X) be a near-continuum of class S 0 .Then there exists δy > 0, δy 0, which is a multiple of δξ , such that ψ(Y) ⊆φ(X) is a near continuum of class S n for all standard n ∈ N, where Y ≡ Zδy and ψ : Y →φ(X) is defined by Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace 17 Proof Applying the Cauchy Principle twice there is > 0, 0 such that δφ(ξ) < for all limited ξ ∈ X.By Robinson's Lemma there is ω +∞ such δy ≡ 1/ω 0; we may suppose that δy is a multiple of δξ .Put Y = Zδy.Observe that if y ∈ Y is limited (5) ψ(y) − y ≤ = δy ω .
Hence ψ(y) − y is an element of the external set of all infinitely large powers of δy, which we denote by £δy ∞ ; this external set is stable by divisions by standard powers of δy.
We end this section by showing that formula (2) holds for the n th order difference quotient of the composite function χ.
to be differentiated discretely, which is, say, of the form We let k = k and we define new θ ij by adding nothing to the θ ij occurring up to the m th factor, and adding 1 to to the θ ij occurring after this factor, with θ n+1j = 0.The term corresponding to the discrete derivative of the m th factor becomes ψ [k ] (φ(ξ)) φ [1] (ξ + θ 1j δξ) φ [2] (ξ + θ 2j δξ)... with and 0 ≤ θ ij ≤ n + 1 for all 1 ≤ j ≤ k i and 1 ≤ i ≤ n + 1.Every product has at most n + 1 factors, and the number of terms in the induction step increases with at most n + 1.In conclusion, the number of terms remains standard finite, each term having at most a standard number of factors.
This operation has the possible effect of regrouping terms with distinct products in some packet of identical powers, but the number of terms resulting from the regrouping does not depend on φ and ψ .Combining, the number of these terms in ( 8) is a function of n and k 1 , ..., k n only, which we may denote by C n,k 1 ,...,kn .

Imme van den Berg
Observing that the coefficients C n,k 1 ,...,kn do not depend on φ and ψ , we choose convenient functions to determine them.As in the proof of the ordinary Faà di Bruno Theorem, we define for k, n ∈ N and a 1 , It follows from the multinomial expansion that for all ξ ∈ X (10) We start with some lemmas.
Lemma 5.10 Let k, n ∈ N be standard with k, n ≥ 1.Then Proof By external induction in n.If n = 1 the results are trivial.Assume the lemma has been proved for n − 1. Observe that where q is a polynomial of degree n − 2 with limited coefficients, containing powers of δξ .Now As long as k < n the induction hypothesis yields δ k−1 ξ n−1 δξ k−1 0 in ξ = 0, and also 0 in ξ = 0, for the coefficients of δ k−1 q(ξ) δξ k−1 will be at most limited.Hence 0 in ξ = 0.If k = n, using the induction hypothesis we find that δ n c δξ n (0) Proof By (10) and Lemma 5.10 non-infinitesimal contributions to δ n c δξ n (0) may occur only for k 1 , k 2 , ..., k n such that k 1 + 2k 2 + ... + nk n = n; in the latter case the coefficient Proposition 5.12 Let n ∈ N be standard and let for k 1 , ..., k n with k 1 +2k 2 +...+nk n = n the constants C n,k 1 ,...,kn be given by Lemma 5.9.Then Proof Because the C(n, k 1 , ..., k n ) of Lemma 5.9 do not depend on the choice of the functions, we choose to determine them for the functions p : X → R, m : Y → R and c = m • p : X → R as defined in (9).Let a 1 , ..., a n ∈ R n be standard.Notice that for 1 ≤ i ≤ n one has p [i] (0) i!a i .Let η = p(0) = 0.Then, with k ≡ k 1 + k 2 + ... + k n , it holds that m [k] (η) = k!.Then it follows from Lemma 5.11 that We define two polynomials u and v in the variables a 1 , ..., a n by Then u(a 1 , ..., a n ) v(a 1 , ..., a n ) for all standard a 1 , ..., a n ∈ R. The polynomial v is clearly standard, and also the polynomial u, because the numbers C n,k 1 ,...,kn are Imme van den Berg standard integers.So u(a 1 , ..., a n ) = v(a 1 , ..., a n ) by the Principle of Carnot.Then u = v by Transfer.This means that their coefficients must be equal.This proves the proposition.
Corollary 5.13 (Infinitesimal Faà di Bruno Theorem).Let n ∈ N be standard.Let φ : X → R and ψ : φ(X) → R be functions of class S n .Then under the conventions mentioned above, for all ξ ∈ [α

Continuisation on near-continua of class S n
We prove the transition from the discrete to the continuous in Theorem 5.1 first of all for the case that n = 1.
Next theorem is formulated for arbitrary standard n ∈ N and presupposes that the conventions at the beginning of Subsection 5.1 hold.Proof By external induction.The case n = 1 is contained in Theorem 5.14.Assume the theorem is proved for n − 1.We follow, mutatis mutandis, the lines of the proof of Theorem 5.14.We prove first that g is of class C n .By Theorem 4.2 the functions f and h are of class C n on [a, b], with f = • (φ [1] ) non-zero, f (m) = • (φ [m] ) for all m with 2 ≤ m ≤ n and h (n) = • (χ [n] ).Observe that f τ n (ξ) = ψ [n] (φ(ξ))(φ [1] (ξ)) n Imme van den Berg change of scale of type centralization-reduction in two dimensions, into the so-called binomial cone.This cone can be seen as the union of two two-dimensional grids, equally spaced in each dimension.Then we will be able to apply the techniques of the previous sections to continuisations on near-continua, and obtain a DeMoivre-Laplace Theorem of higher order: all partial difference quotients of standard order of b(t, x) are nearly equal to the corresponding partial derivatives of G(t, x).
The case of first order in time and second order in space has been proved in [4], using a general method of transition from partial difference equations to partial differential equations of first and second order.With respect the general case of transitions at any standard order we present here a streamlined version of material contained in [11] and [6].
In Subsection 7.1 we effectuate the rescaling of the set of binomial coefficients, recall the nonstandard version of the DeMoivre-Laplace Theorem of [5] and state the higher order DeMoivre-Laplace Theorem.We also discuss the relation of the theorems to some nonstandard approaches to stochastic processes and the heat equation.The higher order DeMoivre-Laplace Theorem will be proved using some partial difference equations for the rescaled binomial coefficients.These will be presented in Subsection 7.2.The transition to continuity towards the higher order DeMoivre-Laplace Theorem will be shown in Subsection 7.3.

2
Imme van den Bergthe function f : the shadow of f being the unique standard real function (if it exists) infinitely close to f on the limited domain.

9 Proposition 3 . 8
Journal of Logic & Analysis 5:6 (2013) Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace Let n ∈ N be standard and f , g : Y → R be functions of class S n .Let a ∈ R be limited.Then f + g and af are of class S n .
) and h is of class C n on [a, b], with for all x ∈ [a, b]

Proposition 5 . 2
Let n ∈ N be standard.Then χ is of class S n on [α • •β] and h is of class C n on [a, b].Proof With external induction.The composition of two functions of class S 0 is clearly of class S 0 .Assume the proposition is proved for n − 1.Let χ = ψ • φ, with ψ and φ of class S n , then in particular χ is of class S 0 on [α • •β].Also φ of class S n−1

Lemma 5 . 3
Let n ∈ N, n ≥ 1 be standard, Y ≡η(X) be a near-continuum of class S n and f : Y → R be a function of class S n .Define g : Y → R by g(η) = f (η + δη).Then g is of class S n−1 in η .

Proposition 5 . 4
Let n ∈ N be standard, Y be a near-continuum of class S n and f , g : Y → R be functions of class S n .Then f • g is of class S n .

Lemma 5 . 5 Proposition 5 . 6
Let n ∈ N be standard, Y be a near-continuum of class S n and f :Y → R be a function of class S n .Let A ⊆ Y be such that A (n) = ∅ and f 0 on the limited part of A.Then 1 f is of class S n on A.Let n ∈ N be standard, Y be a near-continuum of class S n and f , g : Y → R be functions of class S n .Let A ⊆ Y be such that A (n) = ∅ and g 0 on the limited part of A.Then f /g is of class S n on A.Proof By Proposition 5.4 and Lemma 5.5.

Lemma 5 . 11
Let n ∈ N, n ≥ 1 and a 1 , ..., a n ∈ R n be standard.Then Journal of Logic & Analysis 5:6 (2013) Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace 21 By the Theorem of the continuous shadow the function g is of classC 0 on [f (a), f (b)] with h(x) = g(f (x)) for all x ∈ [a, b].Then for all y ∈ [f (a), f (b)] g(y) = h(f −1 (y)).By the usual Chain Rule g is of class C 1 on [f (a), f (b)].Then the Chain Rule may also be applied to the composition h = f • g and yields (12) for all x ∈ [a, b].Let y ∈ [f (a), f (b)] be standard and η ∈ [φ(α) • •φ(β)[ be such that η y.Let ξ ∈ [α • •β[ be such that φ(ξ) = η .Put x = f −1 (y).Then, applying the Theorem of the Journal of Logic & Analysis 5:6 (2013) Discretisations of higher order and the theorems of Faà di Bruno and DeMoivre-Laplace 23

Theorem 5 . 15
Let n ∈ N be standard.Let φ : X → R and ψ : φ(X) → R be functions of class S n .Let ξ ∈ [α • •β].Then g is of class C n on [f (a), f (b)], with g (n) = • ψ [n], and h is of class C n on [a, b], with for all x ∈ [a, b]