CHAP, ill] PARTITIONS. 113 



C. G. J. Jacob! 225 gave fundamental applications of elliptic function 

 formulas to the theory of partitions. He proved the identical relation 



1 + q(v + v- 1 ) + q*(v 2 + v- 2 ) + q g (v* + tr 3 ) + . . 



.-. X (1+ qv)(l + q*v)(l + q 5 v) - 

 X (1 + ger'Xl + gV-'Xl + gHr 1 ) 



if | g | < 1, and another deducible from it by writing qv 2 for v and multiplying 

 by g 1/4 v, viz., 



gl/4(y _j_ y-1) _J_ ^9/4(03 _j_ y-3) _|_ g 25/4(y5 _|_ y-5) _|_ . . . 



. X ? 1/4 (y + v- 1 ) 



From this he inferred, through the intermediary of the four theta functions, 

 the following relations of great importance in the theory of partitions : 



+ 00 



m=l 



= n (i - 9 

 = n 



m=l 



m= oo 

 7T 



7T 



For his expansions, as series in q, of powers of these functions see Chs. VI- 

 IX on sums of squares. If in the first identical relation above we write + z 

 and z in turn for v and multiply the results together, we obtain 



A. M. Legendre 23 noted that Euler's formula (3) implies that every 

 number, not a pentagonal number (3n 2 n)/2, can be partitioned into an 

 even number of distinct integers as often as into an odd number; while 

 (3n 2 db n)/2 can be partitioned into an even number of parts once oftener 

 or once fewer times than into an odd number, according as n is even or odd. 

 This result was implied by Euler 13 ( 46) . 



C. J. Brianchon 24 noted that the literal part of the general term in the 

 expansion of (a\ + a 2 + + a n } m is of the type a? a?, where 

 i + ... + a x = m, x ^ m, x == n. Thus the terms form as many classes 

 as there are values of x, and the terms of a class form as many groups as 

 there are partitions of m into x numbers on. In view of Euler's 9 table we 

 know the number of groups of each class. 



226 Fundamenta Nova Theoriae Func. Ellip., 1829, 182-4. Werke, I, 234-6. Cf. Jacobi. 30 

 See the excellent report by H. J. S. Smith, Report British Assoc. for 1865, 322-75; 

 Coll. Math. Papers, I, 289-94, 316-7. 



23 Th4orie des nombres, ed. 3, 1830, II, 128-133. 



24 Jour, de l'6cole polyt., tome 15, cah. 25, 1837, 166. 



9 



