CHAP, xi] LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES. 337 



Liouville 10 stated that, if &(x, y, z, t) changes sign with x, or y, or both 

 z and t, 



1 ?(2a -2s-l,a-b,a + b,2b + 2s + l) (a 2 + 6 2 = m, a > 0), 



a, b s0 



where the summation on the left relates to the partitions (of any given 

 integer m) 



m = ml + ml + d 3 5 3 (d 3 > 0, 5 3 > 0, 5 3 odd). 



Liouville 11 stated that if $(x, y) is symmetric and even with respect to x, 

 2 (_ i)('-D/2+(d"-D/ 2 ^(^ _ ^', 5' + 5") = S(- IV'-u'VCO, 2d) 



+ 4S(- l)<*- 1 > /2 +< a '- 1 > / V(2di, 2 ai+1 (fe), 



where the summations relate to the partitions, in which m is odd : 

 2m = d'b' + d"5", m = d8, m = di5i + d 2 5 2 , 



all the symbols being positive integers and, with the exception of a 2) odd. 

 In the eighteenth article, Liouville employed a function &(x, y}, odd 

 with respect to x and even with respect to y, and stated that 



2(- iV'-w'Wd' + d", 5' - 5") + ^(d' - d", 5' + ")} 



= 2&(2d, 0) + 4S(- 



For ^"(o;, ?/) = x, the latter gives 



the summations relating to 2m = m' + m", m = mi + 2 at m 2 , where the m's 

 are all odd and positive. 



G. L. Dirichlet 12 proved (a) of Liouville 1 for a = 1. G. Humbert 13 gave 

 a proof by use of infinite series. G. B. Mathews 14 gave a proof. 



J. Liouville 15 stated his 5 formula (7) and that 



SS(- l} m (2 a d + m' - 5)f(2 a d + m', 2m' - 5) = ZS(2 a d - 5)/(ra', 2 a d + 5), 



where the double accents on m, a, d, 5 have been dropped. 



Liouville 16 considered two arbitrary functions /(m) and F(m) having 

 definite values for m = 1, 2, 3, , and set 



where each summation extends over all divisors d of m. For any real or 

 complex numbers ^, v, 



Sd*-'Z,(d)Z l ,() = Sd^Z^dJZ^fi) (5 = wi/d). 



If we take/(m) and F(m) to be powers of m, we obtain a formula concerning 



10 Jour, de Math., (2), 9, 1864, 389-400. Sixteenth article. 



11 Jour, de Math., (2), 10, 1865, 135-144, 169-176 (17th and 18th articles). 



12 Bull, des Sc. Math., (2), 33, I, 1909, 58-60; letter to Liouville, Aug. 27, 1858. 



13 Ibid., (2), 34, 1, 1910, 29-31. 



14 Proc. London Math. Soc., 25, 1893-4, 85-92. 



15 Bull, des Sc. Math., (2), 33, I, 1909, 61-4; letter to Dirichlet, Oct. 21, 1858. 



16 Jour, de Math., (2), 3, 1858, 63-68. 



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