474 Mr WARBURTON, ON SELF-REPEATING SERIES. 



then the fraction which generates the series commencing with C, and extending in the direction 



o 



of the positive indices, will be of higher dimension in the denominator than it is in the nume- 

 rator, by (# + l) units. 



7. Assuming, either first, that A, A, B, and B are finite, and that A , A , , A , 



a b 6-1 6-2 a + 1 



are each equal to zero; or secondly, that C, C , B, and B are finite, and that C , C , , 



b-a Ob -1-2 



C , are each equal to zero ; on either of those assumptions, Equation (6) gives the 



_ [ft- (a + 1)] 



following values of A and of A in terms of the coefficients B and C. 



x a — x 



A = -VB x C + B x.C + ... + B x C ~| ; . (8) 



VB xC +B xC +...+B*C "I; 



L(6-a)fa; -(b-a) (b~a)+l + x -(b + l-a) b -[b-x]J 



A = - VB xC +5 xC + ... +5xC "1. 



a-* U-* -(6-a) 6+1-tf -(6 + 1-a) 6 -[b-a + x]-* 



Both expressions vanish when 00 is either greater than a, or becomes negative. 



(9) 



Sect. IV. On the fraction whose developement is the right arm of a self-repeating 

 recurring series. 



8. Let (A + At + ... + At*+ ... +At a ~ x + ... +At a ~ 1 + At«\ . . (10) 



\0 1 x a-x a-1 a J 



be a rational integer function of t, of the finite dimension, a ; and let a be some integer 

 number or other, constantly odd, or constantly even, for the same function of the form (10). 



Then if there subsists between every two of the coefficients, A, the sum of whose indices is 

 equal to (a), an equation of the form, 



a 



A = (- I) A , . . . . . (11) 



x a—x 



the function itself, and its coefficients, are called recurrent; and we may call equation (11) 



a 



the equation of recurrence, and (- l ) the sign of recurrence, of the function to which they 

 apply. 



When (a) is an even positive integer, and a is an odd positive integer, then, since 



+ A = - A , .-. A = 0. 



a a a 



S 5 2 



When x, in equation (11), is either negative, or greater than a, the corresponding co- 

 efficients will each be equal to zero. 



9. Theorem. Let the series arising from the developement of a proper fraction be the 

 right arm of a self-repeating recurring series. Then, 



First, if the denominator of the fraction is recurrent, the numerator also is recurrent ; and 

 secondly, if the numerator of the fraction is recurrent, the denominator also is recurrent. And, 

 conversely, if the numerator and the denominator of a proper fraction are each recurrent, the 

 developement of the fraction will be the right arm of a self-repeating recurring series. 



