XVIII. On the formulæ investigated by Dr. Brinkley for the general term in the deve­lopment of Lagrange’s expression for the summation of series and for successive integrations

Abstract

In the Transactions of the Royal Society for 1807, Dr. Brinkley has investigated the general value of the coefficient of any term in the development of the function( t / e t -1) n , and his result is remarkable for the mode of its expression in terms of the successive differences of the powers of zero, or of the numbers comprised in the general expression ∆ m 0 n . Since that time, in my paper published in the Transactions of the Society for 1815, “On the Development of Exponential Functions,” I have exhibited other, and much more simple as well as more easily calculable expressions for the same coefficient, by means of the same useful and valuable differences, and in that and other subsequent memoirs, have extended their application to a variety of interesting inquiries in the theory of differences and series. It is singular, however, that up to the present time it has never been shown that the formulæ of Dr. Brinkley, and my own, though affording in all cases coincident numerical results, are analytically reconcileable with each other; nor indeed is it at all easy to see either from the course of his investigation, which turns upon an intricate application of the combinatory analysis, or from the nature of the formula itself, how it is possible to pass from the one form of expression to the other so as to show their identity. This is what I now propose. Referring to my “Collection of Examples in the Calculus of Finite Differences,” will be found the following relation, which enables us to pass from the differences of any one power of zero, as 0 z , to those of any other, as 0 x + n , viz.— {log (1 + ∆) } n . f (∆)0 x = x ( x - 1) .... ( x - n +1). f (∆)0 x - n , or changing x into x + n , {log (1 + ∆)} n . f (∆)0 x + n = ( x +1)( x +2) .... ( x + n ). f (∆)0 x .