Proc. Natl. Acad. Sci. USA Vol. 93, pp. 15004–15008, December 1996 Mathematics
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function (Jacobi continued fractions͞Hankel or Tura´nian determinants͞Fourier series͞Lambert series͞Schur functions)
STEPHEN C. MILNE
Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210
Communicated by Walter Feit, Yale University, New Haven, CT, October 22, 1996 (received for review June 24, 1996)
ABSTRACT In this paper, we give two infinite families of of the constant ZN, which occurs in the evaluation of a certain explicit exact formulas that generalize Jacobi’s (1829) 4 and Epstein zeta function, needed in the study of the stability of -squares identities to 4n2 or 4n(n ؉ 1) squares, respectively, rare gas crystals and in that of the so-called Madelung con 8 without using cusp forms. Our 24 squares identity leads to a stants of ionic salts. different formula for Ramanujan’s tau function (n), when n The s squares problem is to count the number rs(n) of integer is odd. These results arise in the setting of Jacobi elliptic solutions (x1,...,xs) of the Diophantine equation functions, Jacobi continued fractions, Hankel or Tura´nian 2 2 determinants, Fourier series, Lambert series, inclusion͞ x1 ϩ ⅐⅐⅐ϩ xs ϭ n, [1] exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional in which changing the sign or order of the xi’s gives distinct infinite families of identities in this same setting that are solutions. analogous to the -function identities in appendix I of Mac- Diophantus (325–409 A.D.) knew that no integer of the donald’s work [Macdonald, I. G. (1972) Invent. Math. 15, form 4n Ϫ 1 is a sum of two squares. Girard conjectured in 91–143]. A special case of our methods yields a proof of the two 1632 that n is a sum of two squares if and only if all prime conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress divisors q of n with q ϵ 3 (mod 4) occur in n to an even power. in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, Fermat in 1641 gave an ‘‘irrefutable proof’’ of this conjecture. V. & Kac, V. (Birkha¨user Boston, Boston, MA), Vol. 123, pp. Euler gave the first known proof in 1749. Early explicit 415–456] identities involving representing a positive integer formulas for r2(n) were given by Legendre in 1798 and Gauss by sums of 4n2 or 4n(n ؉ 1) triangular numbers, respectively. in 1801. It appears that Diophantus was aware that all positive Our 16 and 24 squares identities were originally obtained via integers are sums of four integral squares. Bachet conjectured multiple basic hypergeometric series, Gustafson’s Cഞ nonter- this result in 1621, and Lagrange gave the first proof in 1770. minating 65 summation theorem, and Andrews’ basic hyper- Jacobi, in his famous Fundamenta Nova (1) of 1829, intro- geometric series proof of Jacobi’s 4 and 8 squares identities. duced elliptic and theta functions, and used them as tools in the We have (elsewhere) applied symmetry and Schur function study of Eq. 1. Motivated by Euler’s work on 4 squares, Jacobi techniques to this original approach to prove the existence of knew that the number rs(n) of integer solutions of Eq. 1 was similar infinite families of sums of squares identities for n2 or also determined by n(n ؉ 1) squares, respectively. Our sums of more than 8 ϱ squares identities are not the same as the formulas of Mathews s n n 3͑0, Ϫq͒ :ϭ 1 ϩ ͑Ϫ1͒ rs͑n͒q , [2] (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, nϭ1 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others. where 3(0, q)isthezϭ0 case of the theta function 3(z, q) in ref. 21 given by
ϱ 1. Introduction j2 3͑0, q͒ :ϭ q . [3] jϭϪϱ In this paper, we announce two infinite families of explicit exact formulas that generalize Jacobi’s (1) 4 and 8 squares Jacobi then used his theory of elliptic and theta functions 2 to derive remarkable identities for the s ϭ 2, 4, 6, 8 cases of identities to 4n or 4n(n ϩ 1) squares, respectively, without s using cusp forms. Our 24 squares identity leads to a different 3(0, Ϫq) . He immediately obtained elegant explicit formulas formula for Ramanujan’s (2) tau function (n), when n is odd. for rs(n), where s ϭ 2, 4, 6, 8. We recall Jacobi’s identities for These results arise in the setting of Jacobi elliptic functions, s ϭ 4 and 8 in the following theorem. Jacobi continued fractions, Hankel or Tura´nian determinants, THEOREM 1.1 (JACOBI). Fourier series, Lambert series, inclusion͞exclusion, Laplace ϱ rqr 4 rϪ1 expansion formula for determinants, and Schur functions. (For 3͑0, Ϫq͒ ϭ 1 Ϫ 8 ͑Ϫ1͒ r this background material, see refs. 1 and 3–16.) rϭ1 1 ϩ q The problem of representing an integer as a sum of squares ϱ of integers has had a long and interesting history, which is ϭ 1 ϩ 8 ͑Ϫ1͒n d qn, [4] surveyed in ref. 17 and chapters 6–9 of ref. 18. The review nϭ1 ͫ d͉n,dϾ0 ͬ 4Թd article (19) presents many questions connected with represen- tations of integers as sums of squares. Direct applications of and sums of squares to lattice point problems and crystallography ϱ r3qr can be found in ref. 20. One such example is the computation ͑0, Ϫq͒8 ϭ 1 ϩ 16 ͑Ϫ1͒r 3 r rϭ1 1 Ϫ q
The publication costs of this article were defrayed in part by page charge ϱ payment. This article must therefore be hereby marked ‘‘advertisement’’ in ϭ 1 ϩ 16 (Ϫ1)dd3 qn. [5] accordance with 18 U.S.C. §1734 solely to indicate this fact. nϭ1ͫd͉n,dϾ0 ͬ 15004 Downloaded by guest on September 28, 2021 Mathematics: Milne Proc. Natl. Acad. Sci. USA 93 (1996) 15005
Consequently, we have where
r ͑n͒ ϭ 8 d and r ͑n͒ ϭ 16 ͑Ϫ1͒nϩdd3, [6] ϱ rsqr 4 8 rϪ1 d͉n,dϾ0 d͉n,dϾ0 U ϵ U (q) :ϭ (Ϫ1) s s r 4Թd rϭ1 1ϩq
respectively. ϱ In general it is true that ϭ (Ϫ1)dϩn͞dds qn nϭ1ͫd͉n,dϾ0 ͬ
r2s͑n͒ ϭ ␦2s͑n͒ ϩ e2s͑n͒, [7] ϭ y1ϩm1 s m1y1 [12] (Ϫ1) m1q . y1,m1Ն1 where ␦2s(n) is a divisor function and e2s(n) is a function of order substantially lower than that of ␦2s(n). If 2s ϭ 2, 4, 6, 8, Analogous to Theorem 1.3, we have Theorem 1.5. then e (n) ϭ 0, and Eq. 7 becomes Jacobi’s formulas for r (n), 2s 2s THEOREM 1.5. including Eq. 6. On the other hand, if 2s Ͼ 8 then e2s(n) is never n 0. The function e2s(n) is the coefficient of q in a suitable ‘‘cusp 16 form.’’ The difficulties of computing Eq. 7, especially the ͑0, Ϫq͒24 ϭ 1 ϩ ͑17G ϩ 8G ϩ 2G ͒ 3 9 3 5 7 nondominate term e2s(n), increase rapidly with 2s. The mod- ular function approach to Eq. 7 and the cusp form e2s(n)is 512 2 discussed in ref. 13. For 2s Ͼ 8, modular function methods such ϩ ͑G3G7 Ϫ G5͒, [13] as those in refs. 22–27, or the more classical elliptic function 9 approach of refs. 28–30, are used to determine general for- where mulas for ␦2s(n) and e2s(n)inEq.7. Explicit, exact examples of Eq. 7 have been worked out for 2 Յ 2s Յ 32. Similarly, explicit ϱ rsqr r formulas for rs(n) have been found for (odd) s Ͻ 32. Alternate, Gs ϵ Gs(q) :ϭ (Ϫ1) r elementary approaches to sums of squares formulas can be rϭ1 1Ϫq
found in refs. 31–36. ϱ We next consider classical analogs of Eqs. 4 and 5 corre- ϭ (Ϫ1)dds qn sponding to the s ϭ 8 and 12 cases of Eq. 7. nϭ1ͫd͉n,dϾ0 ͬ Glaisher (37, 62–64) used elliptic function methods rather ϭ m1 s m1y1 [14] than modular functions to prove the following theorem. (Ϫ1) m1q . y ,m Ն1 THEOREM 1.2 (GLAISHER). 1 1 32 An analysis of Eq. 10b depends upon Ramanujan’s (2) tau 16 ϭ ϩ m1 7 m1y1 [8a] function (n), defined by 3͑0, Ϫq͒ 1 ͑Ϫ1͒ m1q 17 y1,m1Ն1 ϱ 512 q͑q; q͒24 :ϭ (n)qn. [15] 8 2 2 8 ϱ Ϫ q͑q; q) (q ;q͒ , [8b] n 1 17 ϱ ϱ ϭ For example, (1) ϭ 1, (2) ϭϪ24, (3) ϭ 252, (4) ϭϪ1472, where we have (5) ϭ 4830, (6) ϭϪ6048, and (7) ϭϪ16744. Ramanujan r (ref. 2, equation 103) conjectured, and Mordell (38) proved, ͑q; q͒ϱ :ϭ (1Ϫq ). [9] rՆ1 that (n) is multiplicative. In the case where n is an odd integer (in particular an odd n Glaisher took the coefficient of q to obtain r16(n). The same prime), equating Eqs. 10a, 10b, and 13 yields two formulas for formula appears in ref. 13 (equation 7.4.32). (n) that are different from Dyson’s (39) formula. We first To find r24(n), Ramanujan (ref. 2, entry 7, table VI; see also obtain Theorem 1.6. ref. 13, equation 7.4.37) first proved Theorem 1.3. THEOREM 1.6. Let (n) be defined by Eq. 15 and let n be odd. THEOREM 1.3 (RAMANUJAN). Let (q; q) be defined by Eq. 9. ϱ Then Then 1 16 259͑n͒ ϭ ͓17 ⅐ 691 ͑n͒ ϩ 8 ⅐ 691 ͑n) 24 ϭ ϩ m1 11 m1y1 [10a] 3 2 3 5 3͑0, Ϫq͒ 1 ͑Ϫ1͒ m1 q 2 ⅐ 3 691 y1,m1Ն1 ϩ 2 ⅐ 691 ͑n͒ Ϫ 9 ͑n͒] 33152 65536 7 11 Ϫ q(q; q)24 Ϫ q2͑q2;q2͒24. [10b] 691 ϱ 691 ϱ 691 ⅐ 22 nϪ1 Ϫ [†(m)†͑nϪm͒ Ϫ †͑m͒†͑nϪm͒], [16] 2 3 7 5 5 One of the main motivations for this paper was to generalize 3 mϭ1 Theorem 1.1 to 4n2 or 4n(n ϩ 1) squares, respectively, without where using cusp forms such as Eqs. 8b and 10b but still using just sums of products of at most n Lambert series similar to either r † d r [17] r͑n͒ :ϭ d and r(n) :ϭ (Ϫ1) d Eq. 4 or Eq. 5, respectively. This is done in Theorems 2.1 and d͉n,dϾ0 d͉n,dϾ0 2.2 below. Here, we state the n ϭ 2 cases, which determine different formulas for 16 and 24 squares. Remark: We can use Eq. 16 to compute (n)inՅ6nln n steps 2ϩ THEOREM 1.4. when n is an odd integer. This may also be done in n steps by appealing to Euler’s infinite-product-representation algo- 32 rithm (40) applied to (q; q)24 in Eq. 15. (0, Ϫq)16 ϭ 1 Ϫ ͑U ϩ U ϩ U ͒ ϱ 3 3 1 3 5 A different simplification involving a power series formu- lation of Eq. 13 leads to the following theorem. 256 2 THEOREM 1.7. Let (n) be defined by Eq. 15 and let n Ն 3 be ϩ ͑U U Ϫ U ͒, [11] 3 1 5 3 odd. Then Downloaded by guest on September 28, 2021 15006 Mathematics: Milne Proc. Natl. Acad. Sci. USA 93 (1996)
1 where 3(0, Ϫq) is determined by Eq. 3, and Mn,S is the n ϫ n 259͑n͒ ϭ (Ϫ1)dd11 3 matrix whose ith row is 2 d͉n,dϾ0 U ,U ,...,U , if i ʦ S 691 2iϪ1 2(iϩ1)Ϫ1 2(iϩnϪ1)Ϫ1 Ϫ (Ϫ1)dd3(17 ϩ 8d2 ϩ 2d4) [18a] 3 2 and c ,c ,...,c ,if i S, [21] 2 ⅐ 3 d͉n,dϾ0 i iϩ1 iϩnϪ1 ԫ
2 691 ⅐ 2 where U2iϪ1 is determined by Eq. 12, and ci is defined by m1ϩm2 3 Ϫ 2 (Ϫ1) (m1m2) 3 2i m1Ͼm2Ն1 (2 Ϫ 1) m1ϩm2Յn iϪ1 ci :ϭ (Ϫ1) ⅐ ͉B2i͉, for i ϭ 1,2,3,..., [22] gcd(m1,m2)͉n 4i ϫ ͑m2Ϫm2)2 1. [18b] 1 2 with B2i the Bernoulli numbers defined by Eq. 19. y1,y2Ն1 m1y1ϩm2y2ϭn We next have Theorem 2.2. THEOREM 2.2. Let n ϭ 1,2,3,....Then Remark: The inner sum in Eq. 18b counts the number of n 2n solutions (y1, y2) of the classical linear Diophantine equation 4n͑nϩ1͒ nϪp 2n2ϩ3n Ϫ1 3͑0, Ϫq͒ ϭ 1 ϩ (Ϫ1) 2 (r!) m1y1 ϩ m2y2 ϭ n. This relates Eqs. 18a and 18b to the pϭ1 rϭ1 combinatorics in sections 4.6 and 4.7 of ref. 15. In the next section, we present the infinite families of explicit det(Mn,S), [23] ʚSʕIn exact formulas that generalize Theorems 1.1, 1.4, and 1.5. ʈSʈϭp Our methods yield (elsewhere) many additional infinite families of identities analogous to the -function identities in where 3(0, Ϫq) is determined by Eq. 3, and Mn,S is the n ϫ n appendix I of Macdonald’s work (41). A special case of our matrix whose ith row is analysis gives a proof (presented elsewhere) of the two iden- tities conjectured by Kac and Wakimoto (42); these identi- G2iϩ1,G2(iϩ1)ϩ1,...,G2(iϩnϪ1)ϩ1, if i ʦ S n2 or ties involve representing a positive integer by sums of 4 and a ,a ,...,a ,if iԫS, [24] 4n(n ϩ 1) triangular numbers, respectively. The n ϭ 1 case i iϩ1 iϩnϪ1
gives the classical identities of Legendre (ref. 43; see also ref. where G2iϩ1 and ai :ϭ ciϩ1 are determined by Eqs. 14 and 22, 3, equations ii and iii). respectively. Theorems 1.4 and 1.5 were originally obtained via multiple We next use Schur functions s(x1,...,xp) to rewrite The- basic hypergeometric series (44–51) and Gustafson’s* Cᐉ orems 2.1 and 2.2. Let ϭ (1, 2,...,r,...)beapartition nonterminating 65 summation theorem combined with An- of nonnegative integers in decreasing order, 1 Ն 2 Ն ...Ն drews’ (52) basic hypergeometric series proof of Jacobi’s 4 and r...,such that only finitely many of the i are nonzero. The 8 squares identities. We have (elsewhere) applied symmetry length ᐉ() is the number of nonzero parts of . and Schur function techniques to this original approach to Given a partition ϭ (1,...,p) of length Յp, prove the existence of similar infinite families of sums of jϩpϪj squares identities for n2 or n(n ϩ 1) squares, respectively. det͑xi ͒ s x s x ,...,x : [25] Our sums of more than 8 squares identities are not the same ͑ ͒ ϵ ͑ 1 p͒ ϭ pϪj det͑xi ͒ as the formulas of Mathews (31), Glaisher (37, 62–64), Sier- pinski (32), Uspensky (33–35), Bulygin (28, 53), Ramanujan is the Schur function (12) corresponding to the partition . (2), Mordell (26, 54), Hardy (23, 24), Bell (55), Estermann [Here, det(aij) denotes the determinant of a p ϫ p matrix with (56), Rankin (27, 57), Lomadze (25), Walton (58), Walfisz (i, j)th entry aij]. The Schur function s(x) is a symmetric (59), Ananda-Rau (60), van der Pol (61), Kra¨tzel(29, 30), polynomial in x1,...,xpwith nonnegative integer coefficients. Gundlach (22), and Kac and Wakimoto (42). We typically have 1 Յ p Յ n. We use Schur functions in Eq. 25 corresponding to the The 4n2 and 4n(n ؉ 1) Squares Identities partitions and , with .2
r :ϭ ᐉp r Ϫ ᐉ ϩ r Ϫ p To state our identities, we first need the Bernoulli numbers Bn Ϫ ϩ1 1 defined by and r :ϭ jpϪrϩ1 Ϫ j1 ϩ r Ϫ p, t ϱ tn : B , for t Ͻ 2. [19] for r ϭ 1, 2, . . . , p, [26] t ϭ n ͉ ͉ e Ϫ 1 nϭ0 n! where the ᐉr and jr are elements of the sets S and T, with We also use the notation In :ϭ {1,2,..., n}; ʈSʈ is the cardinality of the set S, and det(M) is the determinant of the S :ϭ ͕ᐉ1 Ͻ ᐉ2 Ͻ ⅐⅐⅐ Ͻᐉp͖
n ϫ n matrix M. c The determinant form of the 4n2 squares identity is Theorem and S :ϭ ͕ᐉpϩ1 Ͻ ⅐⅐⅐ Ͻᐉn͖, [27] 2.1. T :ϭ ͕ji Ͻ j2 Ͻ ⅐⅐⅐ Ͻ jp͖ THEOREM 2.1. Let n ϭ 1,2,3,....Then c n 2nϪ1 and T :ϭ ͕jpϩ1 Ͻ ⅐⅐⅐ Ͻ jn͖, [28] ͑0, Ϫq͒4n2 ϭ 1 ϩ (Ϫ1)p22n2ϩn (r!)Ϫ1 3 where Sc : I S is the compliment of the set S. We also have pϭ1 rϭ1 ϭ n Ϫ ͚͑S͒ :ϭ ᐉ ϩ ᐉ ϩ ⅐⅐⅐ ϩ ᐉ det(Mn,S), [20] 1 2 p ʚSʕIn ʈSʈϭp and ͚͑T͒ :ϭ j1 ϩ j2 ϩ ⅐⅐⅐ ϩ jp. [29]
*Gustafson, R. A., Ramanujan International Symposium on Analysis, Keeping in mind Eqs. 25–29, symmetry and skew-symmetry Dec. 26–28, 1987, Pune, India, pp. 187–224. arguments, row and column operations, and the Laplace Downloaded by guest on September 28, 2021 Mathematics: Milne Proc. Natl. Acad. Sci. USA 93 (1996) 15007
4n2 expansion formula (9) for a determinant, we now rewrite then taking q x Ϫq produces the 3(0, Ϫq) in Eq. 20. The Theorem 2.1 as Theorem 2.3. proof of Theorem 2.1 is complete once we show that
THEOREM 2.3. Let n ϭ 1,2,3,....Then 2nϪ1 2 n 2nϪ1 det͓͑sd͞c͒iϩjϪ1͑k ͔͒ ϭ ͑r!͒ 4n2 p 2n2ϩn Ϫ1 rϭ1 3͑0, Ϫq) ϭ 1 ϩ (Ϫ1) 2 (r!͒ pϭ1 rϭ1 and ϫ (Ϫ1)y1ϩ⅐⅐⅐ϩyp 2nϪ1 2 y1,...,ypՆ1 Ϫ͑2n ϩn͒ det͑ciϩjϪ1͒ ϭ 2 ⅐ ͑r!͒. [33] m1Ͼm2Ͼ⅐⅐⅐ϾmpՆ1 rϭ1
m1ϩ⅐⅐⅐ϩmp m1y1ϩ⅐⅐⅐ϩmpyp 2 2 2 ⅐͑Ϫ1) q (mr Ϫ ms) By a classical result of Heilermann (7, 8), more recently 1ՅrϽsՅp presented in ref. 10 (theorem 7.14), Hankel determinants
͚͑S)ϩ͚(T)) whose entries are the coefficients in a formal power series L ⅐ ͑m1m2 ⅐⅐⅐mp) (Ϫ1͒ ⅐det͑DnϪp,Sc,Tc) can be expressed as a certain product of the ‘‘numerator’’ ʚS,TʕIn ʈSʈϭʈTʈϭp coefficients of the associated Jacobi continued fraction J corresponding to L, provided that J exists. Modular transfor- 2ᐉ1ϩ2j1Ϫ4 2 2 2 2 ⅐ ͑m1m2 ⅐⅐⅐mp) s(m1,...,mp)sv(m1,...,mp), [30] mations, followed by row and column operations, reduce the 2 evaluation of det[(sd͞c)iϩjϪ1(k )] in Eq. 33 to applying Hei- c where 3(0, Ϫq) is determined by Eq. 3; the sets S, S ,T, lermann’s formula to Rogers’ (14) J-fraction expansion of the and Tc are given by Eqs. 27 and 28; ⌺(S) and ⌺(T) are given Laplace transform of sd(u, k)cn(u, k). The evaluation of det(c ) can be done similarly, starting with sc(u, k) and the by Eq. 29; the (n Ϫ p) ϫ (n Ϫ p) matrix DnϪp,Sc,Tc :ϭ iϩjϪ1 relation sc(u,0)ϭtan(u). [c(ᐉ ϩj Ϫ1)]1Յr,sՅnϪp, where the ci are determined by Eq. 22, pϩr pϩs The proof of Theorem 2.2 is similar to Theorem 2.1, except with the B2i in Eq. 19; and s and s are the Schur functions in Eq. that we start with sc2(u, k)dn2(u, k). 25, with the partitions and given by Eq. 26. Our proofs of the Kac and Wakimoto conjectures do not We next rewrite Theorem 2.2 as Theorem 2.4. require inclusion͞exclusion, and the analysis involving Schur THEOREM 2.4. Let n ϭ 1,2,3,....Then functions is simpler than in those in Eqs. 30 and 31. We have (elsewhere) written down the n ϭ 3 cases of n 2n 2 Theorems 2.3 and 2.4 which yield explicit formulas for 36 and 4n(nϩ1) nϪp 2n ϩ3n Ϫ1 3͑0, Ϫq) ϭ 1 ϩ (Ϫ1͒ 2 (r!) 48 squares, respectively. pϭ1 rϭ1 ϫ (Ϫ1)m1ϩ⅐⅐⅐ϩmp This work was partially supported by National Security Agency y1,...,ypՆ1 Grant MDA 904-93-H-3032. m1Ͼm2Ͼ⅐⅐⅐ϾmpՆ1 1. Jacobi, C. G. J. (1829) Fundamenta Nova Theoriae Functionum m1y1ϩ⅐⅐⅐ϩmpyp 3 2 2 2 ⅐q (m1m2 ⅐⅐⅐mp) (mr Ϫ ms) Ellipticarum, Regiomonti, Sumptibus fratrum Borntra¨ger;re- 1ՅrϽsՅp printed in Jacobi, C. G. J. (1881–1891) Gesammelte Werke (Rei- mer, Berlin), Vol. 1, pp. 49–239 [reprinted (1969) by Chelsea, ͚(S)ϩ͚(T) ⅐ (Ϫ1) ⅐det͑DnϪp,Sc,Tc) New York; now distributed by Am. Mathematical Soc., Provi- ʚS,TʕIn dence, RI]. S T p ʈ ʈϭʈ ʈϭ 2. Ramanujan, S. (1916) Trans. Cambridge Philos. Soc. 22, 159–184.
2ᐉ1ϩ2j1Ϫ4 2 2 2 2 3. Berndt, B. C. (1991) Ramanujan’s Notebooks, Part III (Springer, ⅐ ͑m1m2 ⅐⅐⅐mp) s(m1,...,mp)sv(m1,...,mp), [31] New York), p. 139. 4. Berndt, B. C. (1993) in CRM Proceedings & Lecture Notes, ed. where the same assumptions hold as in Theorem 2.3, ex- Murty, M. R. (Am. Mathematical Soc., Providence, RI), Vol. 1, cept that the (n Ϫ p) ϫ (n Ϫ p) matrix DnϪp,Sc,Tc :ϭ pp. 1–63. 5. Goulden, I. P. & Jackson, D. M. (1983) Combinatorial Enumer- [a(ᐉ ϩj Ϫ1)]1Յr,sՅnϪp, where the ai :ϭ ciϩ1 are determined by pϩr pϩs ation (Wiley, New York). Eq. 22. 6. Gradshteyn, I. S. and Ryzhik, I. M. (1980) Table of Integrals, We close this section with some comments about the above Series, and Products (Academic, San Diego), 4th Ed.; trans. theorems. To prove Theorem 2.1, we first compare the Fourier Scripta Technica, Inc., ed. Jeffrey, A. and Taylor series expansions of the Jacobi elliptic function 7. Heilermann, J. B. H. (1845) De Transformatione Serierum in f (u, k):ϭsc(u, k)dn(u, k), where k is the modulus. An analysis Fractiones Continuas, Dr. Phil. dissertation (Royal Academy of 1 Mu¨nster). similar to that in refs. 3, 4, and 16 leads to the relation 8. Heilermann, J. B. H. (1846) J. Reine Angew. Math. 33, 174–188. U2mϪ1(Ϫq) ϭ cm ϩ dm, for m ϭ 1,2,3,...,where U2mϪ1(Ϫq) 9. Jacobson, N. (1974) Basic Algebra I (Freeman, San Francisco), and cm are defined by Eqs. 12 and 22, respectively, and dm is pp. 396–397. m 2m 2mϩ1 2 given by dm ϭ [(Ϫ1) z ͞2 ]⅐(sd͞c)m(k ), where z :ϭ 10. Jones, W. B. & Thron, W. J. (1980) in Encyclopedia of Mathe- 2 matics and Its Applications, ed. Rota, G.-C. (Addison–Wesley, 2F1(1͞2, 1͞2; 1; k ) ϭ 2K(k)͞ ϵ 2K͞, with K(k) ϵ K the complete elliptic integral of the first kind in ref. 21, and London), Vol. 11 (now distributed by Cambridge Univ. Press, 2 2mϪ1 Cambridge, U.K.), pp. 244–246. (sd͞c)m(k ) is the coefficient of u ͞(2m Ϫ 1)! in the Taylor 11. Lawden, D. F. (1989) Applied Mathematical Sciences (Springer, series expansion of f1(u, k) about u ϭ 0. New York), Vol. 80. An inclusion͞exclusion argument then reduces the q x Ϫq 12. Macdonald, I. G. (1995) Symmetric Functions and Hall Polyno- case of Eq. 20 to finding suitable product formulas for the n ϫ mials (Oxford Univ. Press, Oxford), 2nd Ed. 13. Rankin, R. A. (1977) Modular Forms and Functions (Cambridge n Hankel determinants det(diϩjϪ1) and det(ciϩjϪ1). Row and column operations immediately imply that Univ. Press, Cambridge, U.K.), pp. 241–244. 14. Rogers, L. J. (1907) Proc. London Math. Soc. (Ser. 2) 4, 72–89. 2n2 n 2n2ϩn 2 15. Stanley, R. P. (1986) Enumerative Combinatorics (Wadsworth & det(diϩjϪ1) ϭ ͑z ͑ Ϫ 1͒ ͞2 ͒det[͑sd͞c͒iϩjϪ1͑k ͒]. [32] Brooks Cole, Belmont, CA), Vol. 1. 16. Zucker, I. J. (1979) SIAM J. Math. Anal. 10, 192–206. 2 From theorem 7.9 of ref. 4, we have z ϭ 3(0, q) , where q ϭ 17. Grosswald, E. (1985) Representations of Integers as Sums of 2 2 exp[ϪK(͌1 Ϫ k )͞K(k)]. Setting z ϭ 3(0, q) in Eq. 32 and Squares (Springer, New York). Downloaded by guest on September 28, 2021 15008 Mathematics: Milne Proc. Natl. Acad. Sci. USA 93 (1996)
18. Dickson, L. E. (1966) History of the Theory of Numbers (Chelsea, Computer Algebra, National Science Foundation Conference New York), Vol. 2. Board of the Mathematical Sciences Regional Conference Series, 19. Taussky, O. (1970) Am. Math. Monthly 77, 805–830. Vol. 66, p. 104. 20. Glasser, M. L. & Zucker, I. J. (1980) Theoretical Chemistry: 41. Macdonald, I. G. (1972) Invent. Math. 15, 91–143. Advances and Perspectives, eds. Eyring, H. & Henderson, D. 42. Kac, V. G. & Wakimoto, M. (1994) in Progress in Mathematics, (Academic, New York), Vol. 5, pp. 67–139. eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. 21. Whittaker, E. T. & Watson, G. N. (1927) A Course of Modern (Birkha¨userBoston, Boston, MA), Vol. 123, pp. 415–456. Analysis (Cambridge Univ. Press, Cambridge, U.K.), 4th Ed., pp. 43. Legendre, A. M. (1828) Traite´des Fonctions Elliptiques et des 464, 498. Inte´grales Euleriennes (Huzard-Courcier, Paris), Vol. III, pp. 22. Gundlach, K.-B. (1978) Glasgow Math. J. 19, 173–197. 133–134. 23. Hardy, G. H. (1918) Proc. Natl. Acad. Sci. USA 4, 189–193. 44. Lilly, G. M. & Milne, S. C. (1993) Constr. Approx. 9, 473–500. 24. Hardy, G. H. (1920) Trans. Am. Math. Soc. 21, 255–284. 45. Milne, S. C. (1985) Adv. Math. 57, 34–70. 25. Lomadze, G. A. (1948) Akad. Nauk Gruzin. SSR Trudy Tbiliss. 46. Milne, S. C. (1987) J. Math. Anal. Appl. 122, 223–256. Mat. Inst. Razmadze 16, 231–275 (in Russian; Georgian summa- 47. Milne, S. C. (1992) Discrete Math. 99, 199–246. ry). 48. Milne, S. C. (1994) SIAM J. Math. Anal. 25, 571–595. 26. Mordell, L. J. (1917) Q. J. Pure Appl. Math. Oxford 48, 93–104. 49. Milne, S. C. (1995) Adv. Math., in press. Acta Arith. 7, 27. Rankin, R. A. (1962) 399–407. 50. Milne, S. C. & Lilly, G. M. (1992) Bull. Am. Math. Soc. (N.S.) 26, 28. Bulygin, V. (1914) Bull. Acad. Imp. Sci. St. Petersbourg Ser. VI 8, 258–263. 389–404; announced by Boulyguine, B. (1914) Comptes Rendus 51. Milne, S. C. & Lilly, G. M. (1995) Discrete Math. 139, 319–346. Paris 158, 328–330. 52. Andrews, G. E. (1974) SIAM Rev. 16, 441–484. 29. Kra¨tzel,E. (1961) Wiss. Z. Friedrich–Schiller–Univ. Jena Math. 53. Bulygin, V. (B. Boulyguine) (1915) Comptes Rendus Paris 161, Naturwiss. Reihe 10, 33–37. 30. Kra¨tzel,E. (1962) Wiss. Z. Friedrich–Schiller–Univ. Jena Math. 28–30 (in French). Naturwiss. Reihe 11, 115–120. 54. Mordell, L. J. (1919) Trans. Cambridge Philos. Soc. 22, 361–372. 31. Mathews, G. B. (1895) Proc. London Math. Soc. 27, 55–60. 55. Bell, E. T. (1919) Bull. Am. Math. Soc. 26, 19–25. 32. Sierpinski, W. (1907) Wiad. Mat. Warszawa 11, 225–231 (in 56. Estermann, T. (1936) Acta Arith. 2, 47–79. Polish). 57. Rankin, R. A. (1945) Proc. Cambridge Philos. Soc. 41, 1–11. 33. Uspensky, J. V. (1913) Commun. Soc. Math. Kharkow (Ser. 2) 14, 58. Walton, J. B. (1949) Duke Math. J. 16, 479–491. 31–64 (in Russian). 59. Walfisz, A. Z. (1952) Uspehi Mat. Nauk (N.S.) (Ser. 6) 52, 91–178 34. Uspensky, J. V. (1925) Bull. Acad. Sci. USSR Leningrad (Ser. 6) (in Russian); English transl., (1956) Am. Math. Soc. Transl. (Ser. 19, 647–662 (in French). 2) 3, 163–248. 35. Uspensky, J. V. (1928) Trans. Am. Math. Soc. 30, 385–404. 60. Ananda-Rau, K. (1954) J. Madras Univ. Sect. B 24, 61–89. 36. Uspensky, J. V. & Heaslet, M. A. (1939) Elementary Number 61. van der Pol, B. (1954) Ned. Akad. Wetensch. Indag. Math. 16, Theory (McGraw–Hill, New York). 349–361. 37. Glaisher, J. W. L. (1907) Q. J. Pure Appl. Math. Oxford 38, 1–62. 62. Glaisher, J. W. L. (1907) Q. J. Pure Appl. Math. Oxford 38, 38. Mordell, L. J. (1917) Proc. Cambridge Philos. Soc. 19, 117–124. 178–236. 39. Dyson, F. J. (1972) Bull. Am. Math. Soc. 78, 635–653. 63. Glaisher, J. W. L. (1907) Q. J. Pure Appl. Math. Oxford 38, 210. 40. Andrews, G. E. (1986) q-Series: Their Development and Applica- 64. Glaisher, J. W. L. (1907) Q. J. Pure Appl. Math. Oxford 38, tion in Analysis, Number Theory, Combinatorics, Physics and 289–351. Downloaded by guest on September 28, 2021