= = U x + (n + 1) ▲ux + (n + 1) n (n+1)n (n-1) ▲3ux+ &c. the form of which shews that the theorem remains true for the next greater value of n, therefore for the value of n still succeeding, and so on ad infinitum. But it is true for n = 1, and therefore for all positive integer values of n whatever. 8. We proceed to demonstrate the same theorem by the method of generating functions. Definition. If (t) is capable of being developed in a series of powers of t, the general term of the expansion being represented by ut, then (t) is said to be the generating function of u. And this relation is expressed in the form 1.2...x is the coefficient of t in the development of e But the first member is obviously equal to GAu, therefore GAu2 = (¦ − 1) $ (t) ....... (3). And generally n GA"ux 1) "4 (1) = (4). To apply these theorems to the problem under consideration we have, supposing still Gu2 = $ (t), Although on account of the extensive use which has been made of the method of generating functions, especially by the older analysts, we have thought it right to illustrate its general principles, it is proper to notice that there exists an objection in point of scientific order to the employment of the method for the demonstration of the direct theorems of the Calculus of Finite Differences; viz. that G is, from its very nature, a symbol of inversion (Diff. Equations, p. 375, 1st Ed.). In applying it, we do not perform a direct and definite operation, but seek the answer to a question, viz. What is that function which, on performing the direct operation of development, produces terms possessing coefficients of a certain form? and this is a question which admits of an infinite variety of answers according to the extent of the development and the kind of indices supposed admissible. Hence the distributive property of the symbol G, as virtually employed in the above example, supposes limitations which are not implied in the mere definition of the symbol. It must be supposed to have reference to the same system of indices in the one member as in the other; and though, such conventions being supplied, it becomes a strict method of proof, its indirect character still remains*. 9. We proceed to the last of the methods referred to in Art. 6, viz. that which is founded upon the study of the ultimate laws of the operations involved. In addition to the symbol ▲, we shall introduce a symbol E to denote the operation of giving to x in a proposed subject function the increment unity;-its definition being Laws and Relations of the symbols E, ▲ and .(2). 1st. The symbol ▲ is distributive in its operation. Thus ▲ (ux + vx + &c.) = Aux + Av2+ &c....... For ▲ (ux+ vx+ &c.) = Ux+1 + Vx+1••• — (Ux + Vx.......) ▲ (ux-v2+ &c...) = ▲ux — Av2+ &c........ (3). 2ndly. The symbol A is commutative with respect to any constant coefficients in the terms of the subject to which it is applied. Thus a being constant, And from this law in combination with the preceding onc, we have, a, b,... being constants, ▲ (aux+bvx+ &c...) = a▲u2 + bAv2+ &c................ .(5). *The student can find instances of the use of Generating Functions in Lacroix, Diff. and Int. Cal. III. 322. Examples of a fourth method, at once elegant and powerful, due originally to Abel, are given in Grunert's Archiv. XVIII. 381. 3rdly. The symbol ▲ obeys the index law expressed by the equation m and n being positive indices. For, by the implied definition of the index m, ▲▲"u, = (▲▲...m times) (AA.....n times) už = {▲▲ ..... (m + n) times} u, = These are the primary laws of combination of the symbol A. It will be seen from these that A combines with A and with constant quantities, as symbols of quantity combine with each other. Thus, (A + a) u denoting Au+au, we should have, in virtue of the first two of the above laws, (A + a) (A + b) u = {A2 + (a + b) ▲ +ab} u =▲3u + (a + b) Au+abu ..... ...... (7), the developed result of the combination (A + a) (A + b) being in form the same as if ▲ were a symbol of quantity. The index law (6) is virtually an expression of the formal consequences of the truth that A denotes an operation which, performed upon any function of x, converts it into another function of x upon which the same operation may be repeated. Perhaps it might with propriety be termed the law of repetition as such it is common to all symbols of operation, except such, if such there be, as so alter the nature of the subject to which they are applied, as to be incapable of repetition*. It was however necessary that it should be distinctly noticed, because it constitutes a part of the formal ground of the general theorems of the calculus. The laws which have been established for the symbol A are even more obviously true for the symbol E. The two symbols are connected by the equation E=1+A, *For instance, if o denote an operation which, when performed on two quantities x, y, gives a single function X, it is an operation incapable of repetion in the sense of the text, since p2 (x, y) = 4 (X) is unmeaning. But if it be taken to represent an operation which when performed on x, y, gives the two functions X, Y, it is capable of repetition since 2 (x, y) = (X, Y), which has a definite meaning. In this case it obeys the index law. B. F. D. 2 founded on the symbolical form of Taylor's theorem. For and from the fact that E and ▲ are thus both expressible by d means of we might have inferred that the symbols E, A, d dx and * combine each with itself, with constant quantities, dx and with each other, as if they were individually symbols of quantity. (Differential Equations, Chapter XVI.) . 10. In the following section these principles will be applied to the demonstration of what may be termed the direct general theorems of the Calculus of Differences. The conditions of their inversion, i.e. of their extension to cases in which symbols of operation occur under negative indices, will d dx * In place of we shall often use the symbol D. The equations will then be E=1+A=e", a form which has the advantage of not assuming that the independent variable has been denoted by x. |