Mathematics. — "Observations on the development of a function 

 in (I .series of factorials'" . I. By Dr. H. B. A". Bockwinkel. 

 (Communicated bj Prof. H. A. Lorkntz). 



(Communicated in the meeting of June 29, 1918). 



1. In his book "Theorie der Garnmafiinktion" N. Nielsen ^ives 

 the necessary and snfticient conditions for the development of a 

 function in a series of factorials. According to him the following 

 proposition holds: 



The necessary and sufficient condition that a function il^x) may 

 be developed in a series of factorials, is that IJ(.i') may be represented 

 as a definite integral of the form 



1 



£Z{x)=^qit){\-~ty ^dt, ..... . (1) 







where rf.{t) is a function with the following properties: 



1. (/{t) is regular within a circle, centi-e ihe origin and with a 

 radius no smaller than unity, so that it may be expaiided in a 

 power-series : 



<f{t) = a, + a,« -h . . . + a,, t" + . . . . . . . . (2) 



2. If 'f^f^t) is the first of the derivatives of '/{t) which is infinite 

 for t = l, then there is a real number / such that 



lim<f.^P){t)^{l—t)-i'^+l>)') (3) 



1) By this notation we mean that lim (l—t)''^'''^^ i.(/0 (0 = or oo, according 



< = i 

 as ?>0 or J < 0. We may express this by saying that -.(/■)(<) is, for t=\, 



equivalent to {l- t)' '''~^l'\ Since -Jj'Kl) is supposed infinite, we have a+P>0. 

 Further it follows from a well-known proposition (Dim, Grundlagen fur eine Theorie 

 der Funktionen einer reellen Grosse, p. 104) that, if -,(1) is itself finite for ^=1, 

 i.e. if p> 1, we must have AH-p< 1. For, if we had X -\- p > 1, then a.''"~^\^) 

 would, according to that proposition, also be infinite, being equivalent to 

 (l—t)~^''^^'~'^\ which is contrary to the hypothesis that ^W')(0 should be ihe first 

 of the derivatives infinite for i=l. We have therefore 



< A + P < 1 . for P > 1 (*) 



from which it follows that A is never positive in this case If, on the other hand, 

 v(/) is itself infinite for ^=1, it is, according to (3), equivalent to (1— O^', and 

 then A is not negative. 



