60 M7' Hardy, On a theorem of Mr G. Polya 



On a theorem of Mr G. Polya. By G. H. Hardy, M.A., 

 Trinity College. 



[Received and read 5 February 1917.] 



1. Mr G. P(51ya has recently discovered a number of very 

 beautiful theorems concerning Taylor's series with integral co- 

 efficients and ' ganzwertige ganze Funktionen '. The latter 

 functions are integral functions which assume integral values for 

 all integral (or for all positive integral) values of the independent 

 variable. One of the most remarkable of these theorems is the 

 following*: • 



Suppose thatg{x) is an integral function, and M{r) the maodmum 

 of I g {x) I for \x\^r. Suppose further that 



g(0),g{l),g{2),... 



are integers, and that 



lim 2-''^JrM{l') = (1). 



?'-»-00 



Then g {x) is a polynomial. 



Mr Polya observes that, if it were possible to get rid of the 

 factor \/r from the equation (1), the theorem could be enunciated 

 in a notably more pregnant form, viz. : 



Among all transcendental integral functions, which assume 

 integral values for all positive integral values of the variable, that 

 of least increase^ is the function 2^. 



Mr Polya states, however, that he has not been able to effect 

 this generalisation. And my object in writing this note is to 

 show that the generalisation desired may be obtained by a slight 

 modification of Mr P(51ya's own argument, and without the 

 addition of any essentially new idea to those which he employs. 



2. Mr Polyaij: reduces the proof of the theorem to a proof 

 that the integral 



T( \— ^' r g(x)dx 



^'^^~2^ij x(x-l)(x-2)...{x-n)' 



extended over the circle \x\ = r - 2n, tends to zero when ?i — » oo , 



*■ G. Polya, ' Uber ganzwertige ganze Funktionen ', Rendiconti del Circolo 

 Matematico di Palermo, vol. 40, 1915, pp. 1—16. See also 'Uber Potenzreihen 

 mit ganzzahligen Koeffizienten ', Mathemathche Annalen, vol. 77, 1916, pp. 497— 

 513, where Mr Polya refers to a third memoir (' Arithmetische Eigenschaften der 

 Eeihenentwicklungen rationaler Funktionen', Journal flir Mathematik) which. 1 

 have not been able to consult, 



t Croissance, Wachstum. 



X Loc. cit., p. 7. 





