ß Art. 4.— M. Kuniyeda : 



4- In Part II, I will give applications of the results, obtained 

 in Part I, to the determination of the behaviour of the coefficients 

 a» of a power series 



00 

 n-O 



asn^oo, whose radius of convergence is unity and which repre- 

 sents a function /C-») having, on the circle of convergence, one 

 singular point only, at z=l. 



As is well known, this problem was first systematically 

 treated by Darboux*. Particularly he considered the case in 

 which /(-^) has a singularity of the type 



f(^) being a function regular for z=l and p denoting any real 

 constant other than zero or a negative integer. His results were 

 extended by Hamyf who considered the case in which f(z) has a 

 singularity of the type 



where ç' is a positive integer. These two authors did not attack 

 the case in which A-?) has an essential singularity for ^=1. This 

 was first done by Fejèr, J who considered the case where 



1 _j_ 



and shewed that 



>/(e7r) 



p being any real constant. 



1 have considered still more general cases in which the func- 

 tion /(^) has a singularity of the following types: 



* Journal de Math. Série 3, t. 4 (1878) pp. 5—57, 377— él7. 



t „ Série 6, t. 4 (1908) pp. 203-283. 



X Comptes JRendm, 30 Nov., 1908 ; and " Äfiympti)tihux értékek megatnrjznsârôl " (1909) 

 Budapest. 



