617 



7' {XV) = xa,v -V- > n a„[ .r — + 7— TTT .... (49) 



1 

 The latter sum may be divided into 



y, (^.«„ -j + ^ (a,, — — j. .... (.50) 

 1 1 



For the first of these two series, with the restriction that the 

 term with n = does not occur, is equal to the series (48) multi- 

 plied bv ./', and therefore it convei-ges, as well as the latter, in the 

 domain {<i) and produces there., together with the term ja^v, the 

 result xTv. The second series is now equally* convergent in («), 

 since the undivided series in (49) converged there; if further A- -|- 1 

 be substituted for n in the second series, the result will be 



T (.rr) = ,r T {v) + ^ a^+i — . 



so that, in connection with (46j, 



X 



r (r) = yt ai+i — (ol) 







This series differs only in so far from (48) that all öt's have moved 

 up one place to the left : from this remark and the fact expressed 

 by (47), that any following derived operation is obtained in the 

 same way from the preceding one ^), the correctness of the repre- 

 sentation (39) for Tm {v) in the domain (<^^) follows at once. 



Let now r(.r) have the radius of convergence r, and let r^ be the 

 «-value to which, for the series P. r corresponds as a i- value. The 

 series P, answering to the transmutation T^ ^ (Tv) is then at all 

 events complete in any domain («) <!^ (r,), with corresponding domain 

 {{i')<^{)'). But the series P^ will often be still complete in a domain 

 greater than {i\), as may be derived from formula (45\ For according 

 to the latter the quantities a',„ are regular in a domain, if this is 

 the case with the transmuted of .r"«r, for arbitrary integral positive 

 values of m, and we amply explained in N". 7 and 8 (second com- 

 munication) that the transmuted Tw of a function ic with radius of 

 convergence r may very well be regular in a circle greater than 



1) We have also to take into account that the formula (.oil holds if r is replaced 

 by XV, which is caused by the fact that it holds for an arbitrary function belonging 

 to (3). ♦ 



