334 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xi 



Liouville 6 stated that, for a function /(a;) = /( x), 



(f) 2(- l)""-5"/(2-"<i" + ) - Zf.On.lrtwO - * J 



where the summations relate to the partitions (2) and m = m\ -f- 2m 2 

 respectively. 



For ra = 8? + 5, f(x) = x sin (xirj2), he derived the relation 



p(m - 4) - 4p(m - 16) + 9p(m - 36) - 



It follows that, if we effect in all possible ways the decompositions 



m = 4s 2 + s* + 4 w = n 2 + 4(X+ ---- hnj) (s>0, n odd and 



Z(- !)<-/ = 2Z(- l)--y. 



If, in place of the second type of decomposition, we employ 



m = r 2 + r! + + rj, 

 where r, ri, , r 4 are positive and odd, then 



4Z(- l) 



For the same two types of partitions and for a function /(x, y), even 

 with respect to x and to y, Liouville stated in his tenth article that 



(77) = or /(m, 2 - 1) + 3/(, 2m - 3) 



+ ... + (2 <<S - !)/( Vm, 1), 



according as m is not or is a square. If f(x, y) is a function of x only, this 

 reduces to (). 



For the same partitions and for a function &(x, y, z, t), even with respect 

 to x, y, z and odd with respect to t, it is stated that 



SS(- l} m "-*&("d" + m', 8" - 2m', 2 a "d" + m' - 5", 5") 



m l 5 2 , 2d 2 2mi 5 2 ) 

 ,j - Vm,j) ( j = 1,3,5, .-, 2^/m- 1), 



3 



according as m is not or is a square. If & = tf(x, y), (?) becomes (77). 

 Other noteworthy cases are & = tf(z) and & = F(f). 



Liouville 7 stated in his eleventh article that, if / is an even function, 



(*) 2Z(- l) (8 "- 1)/2 /(5" - 2m'} = /(l) P (2m - 1) + /(3)p(2m - 9) + -, 

 the summation extending over all integers m' and all divisors 6" of m", where 



(3) m = 2m' 2 + m", m" = d"d" (m" odd and > 0). 



Jour, de Math., (2), 4, 1859, 111-120, 195-204. Ninth and tenth articles. 

 7 Jour, de Math., (2), 4, 1859, 281-304. Eleventh article. 



