DE MOIVRE. 185 



Let Un=l —Vn, and substitute in the equation ; thus 

 The Generating Function of v^ will therefore be 



where (f) is an arbitrary function of t which involves no powers 

 of t higher than f. 



The Generating Function of w„ is therefore 



1 </>© . 



1-^ i-t + hd'r'' 



we may denote this by 



t(0 



{1-t) {\-t + ha''r')' 



where -y^r (t) is an arbitrary function of t which involves no powers 

 of t higher than f^^. Now it is obvious that w„ = if n be less 

 than J), also u^ = a^, and Up^_^ = a^ + ha^. 



Hence we find that 

 so that the Generating Function of u„ is 



(1 - t) {1 - t -{- haH'"-') ' 



The coefficient of f in the expansion of this function will 

 therefore be obtained by expanding 



a^ (1 - at) 



i-t + loFr^ ' 



and taking the coefficients of all the powers of t up to that of 

 f~^ inclusive. 



It may be shewn that De Moivre's result agi'ees with this after 

 allowing for a slight mistake. He says we must divide unity by 

 \—x — ax^ — a^x^ — ... — a^~'^x^, take n—jp + 1 terms of the series, 



multiply by ^^ , and finally put x = j . The mistake here 



