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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n^{2} or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the ηfunction identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n^{2} or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s C_{ℓ} nonterminating _{6}φ_{5} summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n^{2} or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.
 Jacobi continued fractions
 Hankel or Turánian determinants
 Fourier series
 Lambert series
 Schur functions
1. Introduction
In this paper, we announce two infinite families of explicit exact formulas that generalize Jacobi’s (1) 4 and 8 squares identities to 4n^{2} or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s (2) tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. (For this background material, see refs. 1 and 3–16.)
The problem of representing an integer as a sum of squares of integers has had a long and interesting history, which is surveyed in ref. 17 and chapters 6–9 of ref. 18. The review article (19) presents many questions connected with representations of integers as sums of squares. Direct applications of sums of squares to lattice point problems and crystallography can be found in ref. 20. One such example is the computation of the constant Z_{N}, which occurs in the evaluation of a certain Epstein zeta function, needed in the study of the stability of rare gas crystals and in that of the socalled Madelung constants of ionic salts.
The s squares problem is to count the number r_{s}(n) of integer solutions (x_{1}, … , x_{s}) of the Diophantine equation in which changing the sign or order of the x_{i}’s gives distinct solutions.
Diophantus (325–409 A.D.) knew that no integer of the form 4n − 1 is a sum of two squares. Girard conjectured in 1632 that n is a sum of two squares if and only if all prime divisors q of n with q ≡ 3 (mod 4) occur in n to an even power. Fermat in 1641 gave an “irrefutable proof” of this conjecture. Euler gave the first known proof in 1749. Early explicit formulas for r_{2}(n) were given by Legendre in 1798 and Gauss in 1801. It appears that Diophantus was aware that all positive integers are sums of four integral squares. Bachet conjectured this result in 1621, and Lagrange gave the first proof in 1770.
Jacobi, in his famous Fundamenta Nova (1) of 1829, introduced elliptic and theta functions, and used them as tools in the study of Eq. 1. Motivated by Euler’s work on 4 squares, Jacobi knew that the number r_{s}(n) of integer solutions of Eq. 1 was also determined by where ϑ_{3}(0, q) is the z = 0 case of the theta function ϑ_{3}(z, q) in ref. 21 given by Jacobi then used his theory of elliptic and theta functions to derive remarkable identities for the s = 2, 4, 6, 8 cases of ϑ_{3}(0, −q)^{s}. He immediately obtained elegant explicit formulas for r_{s}(n), where s = 2, 4, 6, 8. We recall Jacobi’s identities for s = 4 and 8 in the following theorem.
Theorem 1.1 (Jacobi). and Consequently, we have respectively.
In general it is true that where δ_{2}_{s}(n) is a divisor function and e_{2}_{s}(n) is a function of order substantially lower than that of δ_{2}_{s}(n). If 2s = 2, 4, 6, 8, then e_{2}_{s}(n) = 0, and Eq. 7 becomes Jacobi’s formulas for r_{2}_{s}(n), including Eq. 6. On the other hand, if 2s > 8 then e_{2}_{s}(n) is never 0. The function e_{2}_{s}(n) is the coefficient of q^{n} in a suitable “cusp form.” The difficulties of computing Eq. 7, especially the nondominate term e_{2}_{s}(n), increase rapidly with 2s. The modular function approach to Eq. 7 and the cusp form e_{2}_{s}(n) is discussed in ref. 13. For 2s > 8, modular function methods such as those in refs. 22–27, or the more classical elliptic function approach of refs. 28–30, are used to determine general formulas for δ_{2}_{s}(n) and e_{2}_{s}(n) in Eq. 7. Explicit, exact examples of Eq. 7 have been worked out for 2 ≤ 2s ≤ 32. Similarly, explicit formulas for r_{s}(n) have been found for (odd) s < 32. Alternate, elementary approaches to sums of squares formulas can be found in refs. 31–36.
We next consider classical analogs of Eqs. 4 and 5 corresponding to the s = 8 and 12 cases of Eq. 7.
Glaisher (37, 62–64) used elliptic function methods rather than modular functions to prove the following theorem.
Theorem 1.2 (Glaisher). where we have Glaisher took the coefficient of q^{n} to obtain r_{16}(n). The same formula appears in ref. 13 (equation 7.4.32).
To find r_{24}(n), Ramanujan (ref. 2, entry 7, table VI; see also ref. 13, equation 7.4.37) first proved Theorem 1.3.
Theorem 1.3 (Ramanujan). Let (q; q)_{∞} be defined by Eq. 9. Then One of the main motivations for this paper was to generalize Theorem 1.1 to 4n^{2} or 4n(n + 1) squares, respectively, without 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 Eq. 4 or Eq. 5, respectively. This is done in Theorems 2.1 and 2.2 below. Here, we state the n = 2 cases, which determine different formulas for 16 and 24 squares.
Theorem 1.4. where Analogous to Theorem 1.3, we have Theorem 1.5.
Theorem 1.5. where An analysis of Eq. 10b depends upon Ramanujan’s (2) tau function τ(n), defined by For example, τ(1) = 1, τ(2) = −24, τ(3) = 252, τ(4) = −1472, τ(5) = 4830, τ(6) = −6048, and τ(7) = −16744. Ramanujan (ref. 2, equation 103) conjectured, and Mordell (38) proved, that τ(n) is multiplicative.
In the case where n is an odd integer (in particular an odd prime), equating Eqs. 10a, 10b, and 13 yields two formulas for τ(n) that are different from Dyson’s (39) formula. We first obtain Theorem 1.6.
Theorem 1.6. Let τ(n) be defined by Eq. 15 and let n be odd. Then where Remark: We can use Eq. 16 to compute τ(n) in ≤6n ln n steps when n is an odd integer. This may also be done in n^{2+ɛ} steps by appealing to Euler’s infiniteproductrepresentation algorithm (40) applied to (q; q) in Eq. 15.
A different simplification involving a power series formulation of Eq. 13 leads to the following theorem.
Theorem 1.7. Let τ(n) be defined by Eq. 15 and let n ≥ 3 be odd. Then Remark: The inner sum in Eq. 18b counts the number of solutions (y_{1}, y_{2}) of the classical linear Diophantine equation m_{1}y_{1} + m_{2}y_{2} = n. This relates Eqs. 18a and 18b to the combinatorics in sections 4.6 and 4.7 of ref. 15.
In the next section, we present the infinite families of explicit exact formulas that generalize Theorems 1.1, 1.4, and 1.5.
Our methods yield (elsewhere) many additional infinite families of identities analogous to the ηfunction identities in appendix I of Macdonald’s work (41). A special case of our analysis gives a proof (presented elsewhere) of the two identities conjectured by Kac and Wakimoto (42); these identities involve representing a positive integer by sums of 4n^{2} or 4n(n + 1) triangular numbers, respectively. The n = 1 case gives the classical identities of Legendre (ref. 43; see also ref. 3, equations ii and iii).
Theorems 1.4 and 1.5 were originally obtained via multiple basic hypergeometric series (44–51) and Gustafson’s*C_{ℓ} nonterminating _{6}φ_{5} summation theorem combined with Andrews’ (52) basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n^{2} or n(n + 1) squares, respectively.
Our sums of more than 8 squares identities are not the same as the formulas of Mathews (31), Glaisher (37, 62–64), Sierpinski (32), Uspensky (33–35), Bulygin (28, 53), Ramanujan (2), Mordell (26, 54), Hardy (23, 24), Bell (55), Estermann (56), Rankin (27, 57), Lomadze (25), Walton (58), Walfisz (59), AnandaRau (60), van der Pol (61), Krätzel (29, 30), Gundlach (22), and Kac and Wakimoto (42).
2. The 4n^{2} and 4n(n + 1) Squares Identities
To state our identities, we first need the Bernoulli numbers B_{n} defined by We also use the notation I_{n} := {1, 2, … , n}; ∥S∥ is the cardinality of the set S, and det(M) is the determinant of the n × n matrix M.
The determinant form of the 4n^{2} squares identity is Theorem 2.1.
Theorem 2.1. Let n = 1, 2, 3, … . Then where ϑ_{3}(0, −q) is determined by Eq. 3, and M_{n,S }is the n × n matrix whose ith row is where U_{2i−1 }is determined by Eq. 12, and c_{i }is defined by with B_{2i }the Bernoulli numbers defined by Eq. 19.
We next have Theorem 2.2.
Theorem 2.2. Let n = 1, 2, 3, … . Then where ϑ_{3}(0, −q) is determined by Eq. 3, and M_{n,S }is the n × n matrix whose ith row is where G_{2i+1 }and a_{i} := c_{i+1 }are determined by Eqs. 14 and 22, respectively.
We next use Schur functions s_{λ}(x_{1}, … , x_{p}) to rewrite Theorems 2.1 and 2.2. Let λ = (λ_{1}, λ_{2}, … , λ_{r}, …) be a partition of nonnegative integers in decreasing order, λ_{1} ≥ λ_{2} ≥ … ≥ λ_{r} … , such that only finitely many of the λ_{i} are nonzero. The length ℓ(λ) is the number of nonzero parts of λ.
Given a partition λ = (λ_{1}, … , λ_{p}) of length ≤p, is the Schur function (12) corresponding to the partition λ. [Here, det(a_{ij}) denotes the determinant of a p × p matrix with (i, j)th entry a_{ij}]. The Schur function s_{λ}(x) is a symmetric polynomial in x_{1}, … , x_{p} with nonnegative integer coefficients. We typically have 1 ≤ p ≤ n.
We use Schur functions in Eq. 25 corresponding to the partitions λ and ν, with where the ℓ_{r} and j_{r} are elements of the sets S and T, with where S^{c} := I_{n} − S is the compliment of the set S. We also have Keeping in mind Eqs. 25–29, symmetry and skewsymmetry arguments, row and column operations, and the Laplace expansion formula (9) for a determinant, we now rewrite Theorem 2.1 as Theorem 2.3.
Theorem 2.3. Let n = 1, 2, 3, … . Then where ϑ_{3}(0, −q) is determined by Eq. 3; the sets S, S^{c}, T, and T^{c }are given by Eqs. 27 and 28; Σ(S) and Σ(T) are given by Eq. 29; the (n − p) × (n − p) matrix := []_{1≤r,s≤n−p}, where the c_{i }are determined by Eq. 22, with the B_{2i }in Eq. 19; and s_{λ }and s_{ν }are the Schur functions in Eq. 25, with the partitions λ and ν given by Eq. 26.
We next rewrite Theorem 2.2 as Theorem 2.4.
Theorem 2.4. Let n = 1, 2, 3, … . Then where the same assumptions hold as in Theorem 2.3, except that the (n − p) × (n − p) matrix := []_{1≤r,s≤n−p}, where the a_{i} := c_{i+1 }are determined by Eq. 22.
We close this section with some comments about the above theorems. To prove Theorem 2.1, we first compare the Fourier and Taylor series expansions of the Jacobi elliptic function f_{1}(u, k) := sc(u, k)dn(u, k), where k is the modulus. An analysis similar to that in refs. 3, 4, and 16 leads to the relation U_{2}_{m}_{−1}(−q) = c_{m} + d_{m}, for m = 1, 2, 3, … , where U_{2}_{m}_{−1}(−q) and c_{m} are defined by Eqs. 12 and 22, respectively, and d_{m} is given by d_{m} = [(−1)^{m}z^{2}^{m}/2^{2}^{m}^{+1}]·(sd/c)_{m}(k^{2}), where z := _{2}F_{1}(1/2, 1/2; 1; k^{2}) = 2K(k)/π ≡ 2K/π, with K(k) ≡ K the complete elliptic integral of the first kind in ref. 21, and (sd/c)_{m}(k^{2}) is the coefficient of u^{2}^{m}^{−1}/(2m − 1)! in the Taylor series expansion of f_{1}(u, k) about u = 0.
An inclusion/exclusion argument then reduces the q ↛ −q case of Eq. 20 to finding suitable product formulas for the n × n Hankel determinants det(d_{i}_{+}_{j}_{−1}) and det(c_{i+j−1}). Row and column operations immediately imply that From theorem 7.9 of ref. 4, we have z = ϑ_{3}(0, q)^{2}, where q = exp[−πK()/K(k)]. Setting z = ϑ_{3}(0, q)^{2} in Eq. 32 and then taking q ↛ −q produces the ϑ_{3}(0, −q)^{4n2} in Eq. 20. The proof of Theorem 2.1 is complete once we show that and By a classical result of Heilermann (7, 8), more recently presented in ref. 10 (theorem 7.14), Hankel determinants whose entries are the coefficients in a formal power series L can be expressed as a certain product of the “numerator” coefficients of the associated Jacobi continued fraction J corresponding to L, provided that J exists. Modular transformations, followed by row and column operations, reduce the evaluation of det[(sd/c)_{i}_{+}_{j}_{−1}(k^{2})] in Eq. 33 to applying Heilermann’s formula to Rogers’ (14) Jfraction expansion of the Laplace transform of sd(u, k)cn(u, k). The evaluation of det(c_{i}_{+}_{j}_{−1}) can be done similarly, starting with sc(u, k) and the relation sc(u, 0) = tan(u).
The proof of Theorem 2.2 is similar to Theorem 2.1, except that we start with sc^{2}(u, k)dn^{2}(u, k).
Our proofs of the Kac and Wakimoto conjectures do not require inclusion/exclusion, and the analysis involving Schur functions is simpler than in those in Eqs. 30 and 31.
We have (elsewhere) written down the n = 3 cases of Theorems 2.3 and 2.4 which yield explicit formulas for 36 and 48 squares, respectively.
Acknowledgments
This work was partially supported by National Security Agency Grant MDA 90493H3032.
Footnotes

Walter Feit, Yale University, New Haven, CT

↵ Gustafson, R. A., Ramanujan International Symposium on Analysis, Dec. 26–28, 1987, Pune, India, pp. 187–224.
 Received June 24, 1996.
 Accepted October 22, 1996.
 Copyright © 1996, The National Academy of Sciences of the USA
References
 ↵
Jacobi, C. G. J. (1829) Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti, Sumptibus fratrum Bornträger; reprinted in Jacobi, C. G. J. (1881–1891) Gesammelte Werke (Reimer, Berlin), Vol. 1, pp. 49–239 [reprinted (1969) by Chelsea, New York; now distributed by Am. Mathematical Soc., Providence, RI].
 ↵
 Ramanujan S
 ↵
Berndt, B. C. (1991) Ramanujan’s Notebooks, Part III (Springer, New York), p. 139.
 ↵
 Murty M R
 Berndt B C

 Goulden I P,
 Jackson D M

 Gradshteyn I S,
 Ryzhik I M
 ↵
Heilermann, J. B. H. (1845) De Transformatione Serierum in Fractiones Continuas, Dr. Phil. dissertation (Royal Academy of Münster).
 ↵
 Heilermann J B H
 ↵
 Jacobson N

 Rota GC
 Jones W B,
 Thron W J

 Lawden D F
 ↵
 Macdonald I G

 Rankin R A
 ↵
Rogers, L. J. (1907) Proc. London Math. Soc. (Ser. 2) 4, 72–89.

 Stanley R P
 ↵
 Zucker I J

 Grosswald E

 Dickson L E
 ↵
 Taussky O

 Eyring H,
 Henderson D
 Glasser M L,
 Zucker I J

 Whittaker E T,
 Watson G N
 ↵
 Gundlach KB
 ↵
 Hardy G H
 ↵
 Hardy G H
 ↵
 Lomadze G A
 ↵
 Mordell L J
 ↵
 Rankin R A
 ↵
 Bulygin V
 ↵
 Krätzel E
 ↵
 Krätzel E
 ↵
 Mathews G B
 ↵
 Sierpinski W
 ↵
 Uspensky J V

 Uspensky J V
 ↵
 Uspensky J V
 ↵
 Uspensky J V,
 Heaslet M A
 ↵
 Glaisher J W L
 ↵
 Mordell L J
 ↵
 Dyson F J
 ↵
Andrews, G. E. (1986) qSeries: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, National Science Foundation Conference Board of the Mathematical Sciences Regional Conference Series, Vol. 66, p. 104.
 ↵
 Macdonald I G
 ↵
 Brylinski JL,
 Brylinski R,
 Guillemin V,
 Kac V
 Kac V G,
 Wakimoto M

 Legendre A M
 ↵
 Lilly G M,
 Milne S C

 Milne S C

 Milne S C

 Milne S C

Milne, S. C. (1995) Adv. Math., in press.
 ↵
 ↵
 Andrews G E
 ↵
Bulygin, V. (B. Boulyguine) (1915) Comptes Rendus Paris 161, 28–30 (in French).
 ↵
 Mordell L J
 ↵
 Bell E T
 ↵
 Estermann T
 ↵
 Rankin R A
 ↵
 Walton J B
 ↵
 Walfisz A Z
 ↵
 AnandaRau K
 ↵
 van der Pol B
 ↵
 Glaisher J W L

 Glaisher J W L
 ↵
 Glaisher J W L
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