## New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function (1996)

Citations: | 35 - 1 self |

### BibTeX

@MISC{Milne96newinfinite,

author = {Stephen C. Milne},

title = {New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

Dedicated to the memory of Gian-Carlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the η-function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing

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Citation Context ...ns, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, the Laplace expansion formula for determinants, and Schur functions. This background material is contained in =-=[2, 15, 16, 21, 23, 43, 63, 79, 92, 93, 103, 104, 111, 119, 125, 134, 149, 153, 184, 203, 206, 214, 247, 249, 258]-=-. In order ∑ to make the connection with divisor sums, we follow Jacobi in emphasizing Lambert series ∞ r=1 f(r)qr /(1 − qr ), as defined in [8, Ex. 14, pp. 24], as much as possible. (For Glaisher’s h... |

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Citation Context ... squares of integers is one of the chestnuts of number theory, where new significant contributions occur infrequently over the centuries. The long and interesting history of this topic is surveyed in =-=[11, 12, 26, 43, 94, 101, 102, 188, 200, 204, 229, 241]-=- and chapters 6-9 of [56]. The review article [222] presents many questions connected with representations of integers as sums of squares. Direct applications of sums of squares to lattice point probl... |

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Citation Context ...119, pp. 244–246 ; pp. 223–224], and D. & G. Chudnovsky [47, pp. 140–149]. For the associated continued fraction correspondence we have: Szász [220], Beckenbach, Seidel and Szász [13, pp. 6], Viennot =-=[244]-=-, FlajoletEXACT SUMS OF SQUARES FORMULAS AND JACOBI ELLIPTIC FUNCTIONS 43 [70, pp. 152], Goulden and Jackson [92, Ex. 5.2.21, pp. 312 ; pp. 528–529], Hendriksen and Van Rossum [106, pp. 321], and Zen... |

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Citation Context ...nction and combinatorial formulations of the 6 squares and 6 triangles results in [86, pp. 9–10]. More recent derivations of the 6 squares formulas can be found in Ramanujan [193, Eqns. (135), (136), =-=(145)-=-–(147), Table VI. (entry 1), pp. 158– 159], K. Ananda-Rau [3, pp. 86], Carlitz [37], Grosswald [94, Eqn. (9.19), pp. 121], Hardy and Wright [102, pp. 314–315], Kac and Wakimoto [120, Eqn. (0.3), pp. 4... |

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Citation Context ...s similar to either (1.5) or (1.6), respectively. This condition on the maximal number n of Lambert series factors in each term is consistent with the η-function identities in Appendix I of Macdonald =-=[151]-=-, the two Kac–Wakimoto conjectured identities for triangular numbers in [120, pp. 452], and the above discussion of (1.19). Essentially, expansions of ϑ3(0, −q) N , for N large, into a polynomial of a... |

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Citation Context ...y AMS-TEX2 STEPHEN C. MILNE 1. Introduction In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s =-=[117]-=- 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s [117] explicitly stated degree 2... |

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Citation Context ... 100], Bell [14], Estermann [64], Rankin [200, 201], Lomadze [147], Walton [248], Walfisz [246], Ananda-Rau [3], van der Pol [186], Krätzel [126, 127], Bhaskaran [25], Gundlach [95], Kac and Wakimoto =-=[120]-=-, and Liu [146]. We have found in [164–166] a number of additional new results involving or inspired by Hankel determinants. In [166] we apply the Hankel determinant evaluations in the present paper t... |

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Citation Context ... squares of integers is one of the chestnuts of number theory, where new significant contributions occur infrequently over the centuries. The long and interesting history of this topic is surveyed in =-=[11, 12, 26, 43, 94, 101, 102, 188, 200, 204, 229, 241]-=- and chapters 6-9 of [56]. The review article [222] presents many questions connected with representations of integers as sums of squares. Direct applications of sums of squares to lattice point probl... |

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Citation Context ... be found in [92, pp. 307–308, Ex. 5.2.8, pp. 517–519] and [71]. The sn cn case in (3.8) was first obtained by Ismail and Masson in [111] using a more refined integration-by-parts analysis. Stieltjes =-=[217, 219]-=- first derived (3.5), (3.6), (3.7), and (3.15) by his addition theorem for elliptic functions method. Rogers [206] alsoEXACT SUMS OF SQUARES FORMULAS AND JACOBI ELLIPTIC FUNCTIONS 27 rediscovered the... |

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Citation Context ... is not an integer. The classical formula (1.13) can be rewritten in terms of divisor functions by appealing to the formula for τ(n) from [130, Eqn. (24), pp. 34; and, Eqn. (11.1), pp.36], [140, Eqn. =-=(9)-=-, pp. 111], [8, Ex. 10, pp. 140] given by where τ(n) = 65 756σ11(n) + 691 756σ5(n) − 691 n−1 ∑ 3 σ5(m)σ5(n − m), (1.15) m=1 σr(n) := ∑ d r . (1.16) d|n,d>0 Another useful formula for τ(n) is [140, Eqn... |

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Citation Context ...ting process (computing associated continued fraction expansions), followed by a sieving procedure (inclusion/exclusion). On the other hand, the derivation of the η-function and related identities in =-=[76, 77, 141, 142, 151]-=- relies on first a sieving procedure, followed by taking limits. Theorems 1.5 and 1.6 were originally obtained via multiple basic hypergeometric series [143, 158–161, 163, 168, 169] and Gustafson’s [9... |

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On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field
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Citation Context ...ears that the convergence conditions Re(s) > 5 and Re(s) > 7 are sufficient for Corollaries 7.27 and 7.28, respectively. The analysis involving Theorems 7.24–7.26 may have applications to the work in =-=[32]-=-. 8. The 36 and 48 squares identities In this section we write down the n = 2 and n = 3 cases of Theorems 7.1 and 7.2. These results provide explicit multiple power series formulas for 16, 24, 36, and... |