336 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XI 



according as m is not or is a square, where j = 1, 3, 5, -, 2 Vm 1, 

 m = m' 2 + 2"<T5" (d" t 5" odd and > 0). 



This becomes (e) for F = (- l)-> /2 F(a;, ?/). Next, 



S2F(<T + m 7 , 5" - 2m', 2d" + 2m' - ") 



W 2^TO-l Vm-1 



= or 2 F(i5, s, s) - S *U 2Vm, 20, 



=i '=1 



according as m is not or is a square, the summation on the left relating to 



m = m' 2 + d"" (d" > 0, S" > 0). 

 For T/i, d", 8" odd and positive, 



(x) WF(d" + 2m', 8" - 2m r , 2m f + d" - 5") = (m = 2m' 2 + d"6"). 



Liouville 9 stated that for a function F(x, y, z) odd with respect to x, y, 

 and z, 



(A) ZF(5 3 2m 2 , d 3 + 2m 2 mi, d 3 + 2m2 + mO = 0, 



the summation extending over all partitions of a given integer m = 3 

 (mod 4) : 



m = m\-\- 4ml + 2d 3 5 3 ("h, c? 3 , 5 3 odd, d 3 > 0, 5 3 > 0). 

 Take 



F(x,y,*) = ^^^ ~ * X ' Z - Ji > 



& (x, u) being odd with respect to x, even with respect to u. Then (A) 

 becomes 



(A 2 ) 2<F(5 3 - 2m 2 , d 3 + 2m 2 ) = 2<F(5 3 - 2m 2 , mi). 



With the same notations, Liouville stated in the fourteenth article that 



(B) 2F(5 3 - 2m 2 , d 3 + 2w 2 - mi, 5 3 + mO = 0, 



and if &(x, y, z, t} is changed in sign by a change of sign of x only, or of 

 y only, or of both z and t, 



(C) 2<^"(5 3 2m 2 , d 3 + 2m 2 m\ t d 3 + 2m 2 + mi, 5 3 + Wi) = 0. 



When <^is independent of t or z, (C) becomes (A) or (B), respectively. 

 In the fifteenth article is given the following generalization of (C) : 



5 3 - 2m 2 , d z + 2m 2 - m b c? 3 + 2m 2 + m lt 2 a> 5 3 



a, =0 i 



where a. > 1 and the sign of /3 is chosen so that %(a + /?) is odd, while 

 the summation in the first member applies to the partition 



7^ = 77^ + 4^2 + 2 a3+1 d 3 S 3 (mi, d 3 , 5 3 odd, d s > 0, 5 3 > 0), 

 m being a given odd integer > 1. 



9 Jour, de Math., (2), 9, 1864, 249-256, 281-8, 321-336 (13th-15th articles). 



