116 HlSTOEY OF THE THEOKY OF NtTMBERS. [CHAP. Ill
C. G. J. Jacob!30 stated that if we replace q by qn and set v = =F qm in his first formula,226 we get
(1 d= «w-»)(l =fc gfl+7n)(l - 52n)(l ± 23n-m)(l =fc tf^Xl ~ 24n) • • •
- ft (1 ± g2fn-»-m)(l =£ g2<»-»+™)(l _ ^in) = £ (± l)*0»*+»*.
*=l -00
For m « 1/2, n = 3/2, that with the lower signs becomes Euler's (3). Although he226 (pp. 185-6) gave two simple proofs of it, Jacobi here reproduced Euler's proof in essential points, but with a generalization. He gave a proof of Legendre's23 corollary and proved the following generalization. Let (P, a, 0, • • •) be the excess of the number of partitions of P into an even number of the given distinct elements a, ft • • • , each =f= 0, over the number of partitions into an odd number of them. Then
(P, «, ft %•••)« (P, ft 7, • ' 0 ~ CP ~ «> A % • ' 0-
Let a, ai, • • •, a^\ form any arithmetical progression, and 60, 61, an arithmetical progression with the common difference — a. Set
Then
L s (60, a) + (&i, a, aO + (62, a, ai, a2) = A — (6m, ai, • •
A s [V] - [c0] - [d] + M +
If 60 and a are positive and ma > 60, L vanishes except when fei equals 8i~i + 2s i or 2st-_i + s^ and then equals (— 1)*, where
Si = ai + a2 + • • • 4- a,-.
Jacobi31 noted that Euler9 expressed P = (1 + g)(l + ^2)(1 + 3s) • ' ' in the form /(g2)//(g), where /(a?) 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
Expressing this in the form SJUC^', w^ conclude that, if (7» is the number of partitions of i into arbitrary distinct integers or into equal or distinct odd integers,
where 5 = 1 or 0 according as i is or is not of the form (3n2 d= n)/2. He gave
80 Jour. fOr Math., 32, 1846, 164-175; Werke, 6, 1891, 303-317; Opuscula Math., 1, 1846,
345-356. Cf, Sylvester117, Goldschmidt.1*8 » Jour, fur Math., 37, 1848, 67-73, 233; Werke, 2, 1882. 226-233, 267; Opuscula Math., 2,
1851, 73-80, 113.