(1) (a - a-1) fi (1 - a2x"){\ - a" V')(l - x") = £ (-i)'V. +V"2 + ")/2. «Sl. -oo. Differentiate (1) with respect to a, put a â 1, divide by 2 and we obtain the identity. (2).
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 101. Number 3, November 1987
A SIMPLE PROOF OF JACOBI'S FOUR-SQUARETHEOREM M. D. HIRSCHHORN (Communicated
by Larry J. Goldstein)
Abstract. Jacobi's four-square theorem, which gives the number of representations of a positive integer as a sum of four squares, is shown to follow simply from the triple-product identity.
1. Recently [2], I showed how one can obtain Jacobi's two-square theorem from the triple-product identity. In this note I show that the triple-product identity also gives Jacobi's four-square theorem: Theorem 1. The number rA(n) of representations of four squares is given by
r4(«) = 8
£
of the positive integer n as a sum
d.
l
x4^^-
n (i + ^2"-i)2(i -x2")) dx „Vf yj
Employing the product rule to evaluate the derivatives, we find
non» 1
x"t = n (i + *2"-')2(i - *2")(i + *2")2(i - x2") n» 1
L
. ~ —
\
2«jc2" i
i
2«
n » 1 x + ■*
. v "
2«x2" i _
2n
w» 1 1
■*
,
- n o + *2")2(i - *a")o + *2"-i)2(i - x2") n>l
x|>£ (1> 1
("-')"';-'-4£ 1 _|_
1 + A
2n-l
w —
1 _
»> 1 1
x
2((
or,
noí¡> i
*")6 = n o + x2"_i)2(i + *2")2(i - x2")2 «>i x(l-8El(2W"1)x2"" 1 + x2""1
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^ 1 + x2"
438
M. D. HIRSCHHORN
Dividing both sides by
n o + x")\i - x"f = n o + *")2(i- x2")2
n Ss 1
n> 1
= n
a + x2""1)2^
+ *2")2(i -x2")2,
n»l
we obtain
I1"*")4-1
8£^(2w"1^2'"'
„Vi U + x"}
X\ \
2nx2r
1 + *2"-1
1 + x2"
Now, it is a simple consequence of the triple-product identity that
n (££) -£(-i)v. so we have
- 00
/
1 + x2""1
(I » 1
1 + x2n
Putting -x for x, we obtain
£,-),.i+«s( 1 1I
1 - X2"
1 + X2"
4nx4"
n>l *■~ x
or.
nx
£*"* =i + 8£r i from which Theorem
1 follows directly.
References 1. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon
Press,
Oxford. 2. Michael
D. Hirschhorn,
A simple proof of Jacobi's
two-square theorem. Amer. Math. Monthly 92
(1985), 579-580. School of Mathematics, University South Wales, Australia 2033
of New South Wales, P. O. Box 1, Kensington,
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