188 



(1). When n>m (o) slinws- tho ratio to bo prrrator than 1 and therefore 

 the series S(x) diverges. 



(2). When n<m (5) shows that for increasintt t the ratio approaches 

 zero and therefore the series S(x) converges. 



(3). When n = m we see from (6) that the convergence of the .series de- 

 pends on the value of |a'. 



If ja|>l the ratio ultimately approaches a quantity less than 1 and 

 therefore S(x) converges. 



If |a|<l the ratio is greater than 1 and S(x) diverges. 



If |a| = l the series will converge when* k — 1>1. 



In the cases where S(x) converges, except where n = m and |a]=l, the 

 ratio ut+i 'ut has been shown to approach a quantity which is less than 

 1 for every x in T. Hence an M and an r can be found such that 



M+Mr+Mr2 +Mr3 +Mr^ + 



is a convergent series of positive constant terms which is greater term by 

 term than the series 



(7) Uu + Ui + U> + U3+ 



for every x in T. Therefore the series S(x) converges uniforndy in T and is 

 an analytic function in that region since each term is analytic in T. In case 

 n = m and |a| =1 we see from (6) that the coefficient of 1/t does not contain 

 X but that the coefficients of higher powers of 1 't do. These coeflicients are 

 polynomials in x. If we replace each x by a cjuantity which is greater 

 than the greatest absolute value of x in T and replace the coefficients of the 

 powers of x by their absolute values, then the ratio (Ctj is increased but is 

 still such that a .series of positive constants can be constructed which is con- 

 vergent and is term by t<M'm greater than the series (7). Hence S(x) con- 

 verges uniformly in T when n = m and |a|=l and is therefore analytii in T. 

 But T is any closed region in the strip and hence S(x) is analytic throughout 

 the strip. 



Under the conditions of the theorem Sfx) has been s!i()\vn to be a solution 

 of (he difference ecjuation of the theorem with the recpiired properties I and 

 11. The general solution having those properties will be obtained by adding 

 to thi^ j)artii'ulai' solution the genei'al solution of the homogen(M)Us (>(]uation 

 as found in the theorem of §2 wliich lias t lie same nropcities. This com- 

 pletes the theorem. 



*In :i?c )rd in -o willi :i I lie )rcm of (! im-i. Se,' ().)cra. vol. :!, p. l.'iO. 



HI (mill iiK/lmi . f ml ill 11(1 . 



