Empirical Expansions of Modular Functions 119 



Taking .9=1, pAp = Qp", 



so that -Agp^ AsAp... (3). 



If no restrictions are placed on s we find, by equating coefficients 

 of rP', 



^^s+^,Asp.= QAsp. 



From this 



Asp.-ApA,p + ^p''-'A. = (4). 



From equations (3) and (4), we can prove that A^n = A^An 

 if m and n are prime to each other. For all we really have to shew 

 is that, if 2? is a prime and s is prime to j), then ^,,^a = AifAp\. 

 But from equation (4), we have 



Asp\+2 - ApAspK-hi + ^p"-^ Agp\ = 0, 



and Ap\+2 — ApAp\+i + ^j)''-^ A^k = (4a). 



Hence the theorem follows by induction, for if it is true for \ 

 and X + 1 it is true for X, + 2. But it is true for X = and for 

 X= 1 (equation 3): hence it holds universally. 



We notice also that equation (4a) is a linear difference equation 

 of the second order with constant coefficients*. Hence, since 

 A,= l, 



1 + ApX + Ap-iX" 4 Ap^x^ + . . . = 1/(1 - ApX + ^p^-Kx-), 

 from which, by putting x = l/jf, 



pS p2S p3S '^ ••■ / ^ p' p^^ J' 



Putting for p in succession the primes 2, 3, 5 ..., multiplying 

 together the corresponding equations, and remembering that 

 A,nn = A^nAn if vi and /; are prime to each other, we have 



« 



where the product refers to the primes 2, 3, 5 — 



The simplest application of these results is given by the 

 function 



fa {co„ &).3) = A ^— ft),, (Ooj 



This is obvious if we put fx^= ApK. 



