Empirical Expansions of Modular Functions 123 



where [- j and ( ] are symbols of quadratic reciprocity, so 



that (:-^) =(-l)V , (I) = 1 if i, EE ± 1 (mod 12), and (|) = - 1 



if jj = ± 5 (mod 12). If /; = -S, ('^\ = 0. 



These are particular cases of Euler's theorem that 



s-^^^^=ni/(i--^^^) 



if the function y" satisfies the condition 



f{m7i)=f(m)f(n), 



the product refers to any group of primes, and the summation to 

 all numbers whose prime factors are included in the group. Thus 



r (1 _ r2J)(l _ r'') ... = i (_i)»y.(««+i)^= v f'}\ ,,.«^ . 



-c»' 1,3, 5... V''^/ 



and r [(1 - ?•») (1 - r^^) ...]'= t (- 1) ^ wr^' = 5 f ^ nr"\ 



1,3,5... 1,3,5... \ n J 



Finally, Mr Raman ujan gives two results, equations (155) and 

 (162), of which the first is 



5 -^ = lT9i=^ n 1/(1 - 2c, p-' + (- IV P'-n 



where Cp = u^ - (4w)- and u and y are the positive integers satis- 

 fying u^ H- {^v)'=}f: But if JO = 3 (mod 4), Cp is taken to be zero, 

 /lo (w) is defined * by 



=^r[(l-r^)(l-r^)(l-r«)...]"/[(l+r)(l-r-^)(l-|-r^)(i-r^)...h 

 and this is equal tof 



ii 2(a,- + i2/)^r^'+^'. 



— 00 —00 



The second result is 



I'^^^lT^^W-Sc^p-^+i'^-'X (1^ = 3,5...), 



" The functions /io(?i),/i6 (n) arise in iinding the number of representations of 

 n as a sum of 10 and 16 squares respectively and the series 2 S (a; + i?/)-*r^""'"*' is 

 well known in this connection. 



t From this, it follows that the result can be also proved as a particular case 

 of Euler's product. 



