772 



in (1) R{X)<C_1 was necessary, so we find apart from other condi- 

 lions, tliat (2) is a solution of (1), provided < 7^(/) < 1. 



2. If we do not impose tliat great restriction on ft in (9) (viz. 

 ft = /), bul if we maintain its independence in reference to /, (9) 

 even in this case, apart from other conditions, represents a solution 

 (1) provided /?(?.) < 1 is satisfied. 



A simpler forin may be given to (9) t)y using the following 

 definition : 



Dl icl' — "-J—--!— ieP-9 



r{p-g + l) 



for all values of p and q, which form exactly indicates for positive 

 integer values of </ the 7''' differential quotient of xi'. For the series 

 occurring in the second member of (9) under the integral sign, we 

 may then write: 



2: (§— a)'"-/^ = 



. r(,„,_|_i_„) ^* ' . 



(-i)"'«™(0" 



,", r(m+2-T) ^" ' r(i— ;.) 



= A''+'-^ ^ — : — ^"4^ (i-«)"'+-' = ---^, n'?'^' ym 



on account of (7), or again =— — r /'^'''+'~'' (s) according to the 



1 (I /) 



well-known notation by the whole index. 



So we get for (9) the form : 



„(^.) := ±^ r{X) r^'y~'j(..-§r V 7_^j_^. (i, i/,^) ƒ r,.+.-;XD^S,(l 1) 



a 



0<fl(fi)<« + 1. 



We now recognize at once (2) from (11) for ft := A. 



The remaining conditions, under which (11) will be a solution of 

 (1), are implied in the way of deducing these relations from (5) 

 and (6), in which we liave carried out summations under the integral 

 sign. 



So it is necessary that in (11) the series for /i/'+i-') (.t), cf. (10), 

 converges uniformly for a^.v^b; if this condition is satisfied, the 

 series (8) for u{x) is uniforndy convergent, as in the second member 



