XX. On Self-repeating Series. By Henry Warburton, M.A. 



[Read May 15, 1854.] 



Sect. I. Introduction. 



1. On reading some years ago the well-known Memoir of Laplace, in which he institutes 



a comparison between the general terms of the exponential functions — — and ■ , I noticed 



a circumstance not mentioned by the distinguished Author, and, as I conceive, overlooked 

 by him. 



The general term of is ["0* - 1* + 2* - 3 X + &c.l; and we are to substitute for 



1 + e* 1.2...a> L J 



the infinite series which this term contains, its generating fraction. On computing the gene- 

 rating fractions of the series (l* - 2 x t + 3 x t* - 4?f + &c.), when the integer values 1,8, 3, &c. 

 were successively assigned to a?, I found that the numerators of these fractions, so far as my 

 computations extended, were all recurrent functions of t. Thus, for instance, 



when m m l, the numerator = 1, 



= 2, = 1-t, 



= 3, = 1 -U+ t\ 



= 4, = 1 - lit + Uf-t 3 , 



= 5 = 1 - 26t + 66f - 26t* + t\ 



And so on ; and if in the infinite series, the terms, retaining the same numerical values, 

 were made all positive, then the terms of the several numerators, retaining the same numerical 

 values, became all positive; and it therefore appeared that the generating fraction of the 

 infinite series (l* + 2,'t + 3 X C + &c.) had a recurrent numerator. 



Dr Brinkley and Sir John Herschel, who have followed up with great success the 

 researches of Laplace, have considered with particular attention the series (l*— 2*t + 3 x t % — &c); 

 but they make no allusion to the fact of the fractions which generate the different series of this 

 form, when different consecutive integer values are given to x, having recurrent numerators. 



I was therefore led to investigate the question, what are the conditions which the denomi- 

 nator of the generating fraction, and the terms of the recurring series generated must satisfy, in 

 order that the numerator of the generating fraction may be a recurrent function of t. 



It is the result of that investigation which I now communicate to the Society. 

 Vol. IX. Part IV. 61 



