XXII. Consideration of various points of analysis

Abstract

On the Calculus of Generating Functions. In whatever point of view we consider the theory of generating functions, whether as the fertile source of new discoveries, or as a medium for exhibiting, in the most comprehensive and uniform point of view, results already known; we shall find fresh cause to admire the profound and original genius of its author. To the latter of these objects it is, however, more peculiarly adapted, and perhaps, in the present state of analytics, this may even be considered as the more precious advantage of the two. Such has been the indefatigable activity of those illustrious men, who have devoted themselves to the pursuit of mathematical science, that analysis must be considered as already adequate to every purpose to which we can reasonably hope to see it applied. The attention of the scientific observer must now be directed to those elevated stations, from which distinct and extended views of its arrangement as a whole can be obtained. The calculus of generating functions affords such a station, and commands a wider and more magnificent prospect than any which has yet been opened to the view of the speculative philosopher. It becomes interesting then to extend its application as far as possible in this line, and to introduce it on every occasion where there seems any probability of its coming successfully into play. Such have been in part the considerations which determined me to adopt it as a vehicle in laying before the Society some results of a singular and interesting nature, derived indeed originally from other principles, but which, like all the rest, flow with the utmost facility from the first elements of this calculus. In the following pages I have uniformly made use of the functional or characteristic notation; together with the method of separating ( where it could conveniently be done) the symbols of operation from those of quantity. This method I have, perhaps, extended and carried somewhat farther than has hitherto been customary; but, I trust, without losing sight of its grand and ultimate object, the union of extreme generality with conciseness of expression. To avoid the necessity of continual explanation, I shall here set down the leading points of the system.