116 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



C. G. J. Jacob! 30 stated that if we replace q by q n and set v = T q m in 

 his first formula, 226 we get 



q n ~ m )(l q n+m }(l - g 2 ")(l =b 



= ft (1 q^n-n- 



For m = 1/2, ?i = 3/2, that with the lower signs becomes Euler's (3). 

 Although he 226 (pp. 185-6) gave two simple proofs of it, Jacobi here repro- 

 duced Euler's proof in essential points, but with a generalization. He 

 gave a proof of Legendre's 23 corollary and proved the following generaliza- 

 tion. Let (P, a, j8, ) be the excess of the number of partitions of P 

 into an even number of the given distinct elements a, /?, , each =j= 0, 

 over the number of partitions into an odd number of them. Then 



(P, a, ft 7, - (P, ft 7, - (P'- ", ft 7, ' ) 



Let a, i, , a m _i form any arithmetical progression, and 6 , 61, , b m 

 an arithmetical progression with the common difference a. Set 



d = 

 Then 



L = (6 , a) + (61, a, fli) + (6 2 , a, Oi, a 2 ) + + (&-i, a, a\, '> a m -i) 

 = A 



, m- 



A = [60] - M - [cj + [dj + 



If 6 and a are positive and ma > b , L vanishes except when 61 equals 

 s_i + 2si or 2sf_i + S{ } and then equals ( I) 1 , where 



Si = Oil + #2 + ' + Ctf. 



Jacobi 31 noted that Euler 9 expressed P = (1 + g)(l + g 2 )(l + g 3 ) 

 in the form /(<? 2 ) //(<?), where /(re) is given by (3). Jacobi expressed P in 

 six ways as quotients of two infinite products and expanded each into 

 infinite series ; the next to the last case is 



- g 9 ) 2 (l - g 



Expressing this in the form 2* =1 C^ y , we conclude that, if d is the number 

 of partitions of i into arbitrary distinct integers or into equal or distinct 

 odd integers, 



Ci = 2 { Ci-3 C,'_12 + C;_27 C,_48 + Ci-75 ~ ' ' ' } +5, 



where 5 = 1 or according as i is or is not of the form (3n 2 n)/2. He gave 



30 Jour, fur Math., 32, 1846, 164-175; Werke, 6, 1891, 303-317; Opuscula Math., 1, 1846, 



345-356. Cf. Sylvester 117 , Goldschmidt. 148 



31 Jour, fur Math., 37, 1848, 67-73, 233; Werke, 2, 1882, 226-233, 267; Opuscula Math., 2, 



1851, 73-80, 113. 



