104 HISTORY OP THE THEORY OF NUMBERS. [CHAP, ill 



L. Euler 9 noted that the coefficient of x n z m in the expansion of 



is the number of different ways n is a sum of m different terms of the set 

 a, |8, 7, . The coefficient of x n z m in the series giving the expansion of 



is the number of different ways n is a sum of m terms of ex, /?, , repetitions 

 allowed. In particular, the coefficient of x n in 



is the number of partitions of n. If the product extends only to j = m, the 

 coefficient of x n is the number of partitions of n into parts ^ m. In 



z = U 



replace z by xz, so that Z becomes Z/(l + xz). Hence 

 (1 + xz)(l + PIXZ -f 



By comparison of coefficients, we get 



i 

 P 



* m ' 



(1 - z)(l - z 2 ) --(I - a;" 1 )' 



Hence the number of partitions of n into parts ^ m equals the number of 

 ways of expressing n + m(m + l)/2 as a sum of m distinct parts. Applying 

 the same process to n(l x j z)~ l , we obtain the series 



, , __ __ __ __ 



" ~ ~ ~ ' ^z 2 ) " " (T- - ~ 



Hence the number of partitions of n into parts ^ m equals the number of 

 ways of expressing n + m as a sum of m parts, not necessarily distinct. 

 If (n, m) is the number of partitions of n into parts ^ m, then 



(n, m) = (n, m 1) + (n m, m). 



By use of this recursion formula, Euler computed a table of the values of 

 (n, m) for h ^ 69, m ^ 11. The product of 



P = ft (1 - arO, Q = ft (1 + 



is 11(1 x 2i ), all of whose factors occur in P. Hence [proof by L. 

 Kronecker 10 for | x \ < 1, to insure absolute convergency^], 



PQ 1 



(4) 



-aO(l -z 3 )(l - 



9 Introductio in analysin infinitorum, Lausanne, 1, 1748, Cap. 16, 253-275. German transl. 



by J. A. C. Michelsen, Berlin, 1788-90. French tranel. by J. B. Labey, Paris, 1, 1835, 

 234-256. 



10 Vorlesungen liber Zahlentheoric, 1, 1901, 50-56. 



