188 



(1). When n>m (5) shows the ratio to be greater than 1 and therefore 

 the series S(x) diverges. 



(2). When n<m (5) shows that for increasing 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 |a|>l the ratio ultimatelj' approaches a quantit\- 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+Mr^' +Mr-' +Mv* + 



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

 term than tlie series 



(7) Uo + U, + U: + U.-.+ 



for every x in T. Therefore the series .S(x) converges uniformly' 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 coefficients are 

 polynomials in x. If we replace each x by a ciuantitj* 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 (0) is increased but is 

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

 vergent and is term by form greater than the series (7). Hence .S(x) con- 

 verges uniformK' in T when n = m and |a|=l and is therefore analytic in T. 

 But T is any closed region in the strip and hence S(x) is anal5'tic throughout 

 the strip. 



Under the conditions of the theorem S(x) has been shown to be a solution 

 of the difference equation of the theorem with the required properties I and 

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

 to this particular solution the general solution of the homogeneous equation 

 as found in the theorem of §2 which has the same projierties. This com- 

 pletes the theorem. 



*In a?c>r(lin-e with a ihorein of fj im-i. Se^ O.)ora, vol. Z, p. 139. 



Bloom higlon , Tndiiuiti . 



