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 4n2 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 η-function 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 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 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 n2 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.

Magnesium-Chelatase from Developing Pea Leaves1 : Characterization of a Soluble Extract from Chloroplasts and Resolution into Three Required Protein Fractions

Mg-chelatase catalyzes the ATP-dependent insertion of Mg2+ into protoporphyrin-IX to form Mg-protoporphyrin-IX. This is the first step unique to chlorophyll synthesis, and it lies at the branch point for porphyrin utilization; the other branch leads to heme. Using the stromal fraction of pea (Pisum sativum L. cv Spring) chloroplasts, we have prepared Mg-chelatase in a highly active (1000 pmol 30 min−1 mg−1) and stable form. The reaction had a lag in the time course, which was overcome by preincubation with ATP. The concentration curves for ATP and Mg2+ were sigmoidal, with apparent Km values for Mg2+ and ATP of 14.3 and 0.35 mm, respectively. The Km for deuteroporphyrin was 8 nm. This Km is 300 times lower than the published porphyrin Km for ferrochelatase. The soluble extract was separated into three fractions by chromatography on blue agarose, followed by size-selective centrifugal ultrafiltration of the column flow-through. All three fractions were required for activity, clearly demonstrating that the plant Mg-chelatase requires at least three protein components. Additionally, only two of the components were required for activation; both were contained in the flow-through from the blue-agarose column.