Mr Hardy, On a theorem of Mr G. Polya 63 



The proof of the theorem conjectured by Mr Polya is thus 

 completed. 



5. Mr Polya has also proved an analogous theorem concerning 

 integral functions which assume integral values for all integral 

 values of x, viz.: 



If ...,g{-2), g{-l), g{0\ g{\), g{2)... 



are integers, and 



lim (^^rVrilf(r)=0 (3), 



then g {a;) is a polynomial. 



His proof applies, as it stands, to odd functions only, its appli- 

 cation to a completely general function demanding the more 

 stringent condition 



lim 



r-*-co 



^4r^l 'rm(r) = (3'). 



He states that it is possible to replace the index | by ^ in all 

 cases, but that, as he has not been able to reduce the condition to 



lim (^^^y*'ilf(r) = (3"), 



he has not thought it worth while to publish the details of his 

 work. 



A modification of Mr P<51ya's argument, in every way similar 

 to that which I have made in the proof of his first theorem, 

 enables us to replace (3) by (3") when g{x) is odd. The same 

 modification in his unpublished argument would, I presume, be 

 equally effective in general. 



That the number 



3 + ^/5 



cannot be replaced by any larger number, and so really is the 

 number which ought to occur in any theorem of this character, 

 is shown by Mr Polya by the example of the function 



which assumes integral values for all integral values of .r. 



