Mathematics. — “On a certain point concerning the generating 
functions of Larrace.” By Dr. H. B. A. BockwinkeL. (Com- 
municated by Prof. H. A. Lorentz). - 
(Communicated in the meeting of May 31, 1919). 
1. The following remarkable proposition of the integral fer p(r)dr, 
0 
or of the integral 
1 
aldi Ld de, oneens vavo BED 
J 
derived from the former by the substitution r= —log t‚ has been 
proved by Lercu’): 
If the determining function a(x) vanishes for an arithmetical 
progression of values of « with positive common difference u 
=d uy, D= © Bad Wy PAAR bel A el gs pte 
then it vanishes for all values of x, and the generating function f(t) 
also vanishes. 
Leren uses for the proof a theorem of Wererstrass, according to 
which any function which is continuous in a closed interval can be 
represented by a uniformly converging series of rational integral 
functions. Since the theorem, which is also mentioned by PINCHERLE *) 
and by Nirisen’®), has a great many interesting consequences, it 
seems not unuseful to prove it in a manner which is independent 
of Werersrrass’s theorem. The reasoning we give in the next pages 
makes use of the theorem of FourIEr. 
2. The following suppositions are sufficient for the purpose: 
1. The function f(t) is continuous in the interval of integration, 
with possible exception as to the value ¢= 0. 
1) Acta mathem. 27, 1908. 
8) “Sur les fonctions déterminantes’’, Ann. de 'Éc. Norm. 22, 1905. PincHErRte 
calls f(t) “fonction génératrice’ and x (x) “fonction determinante’, whereas Leren 
does the reverse. We have followed the nomenclature of PincHERLE in the text. 
3) “Handbuch der Gammafunktion"’, p. 118. 
