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.

A Cross-Polarization, Magic-Angle-Spinning, 13C-Nuclear-Magnetic-Resonance Study of Polysaccharides in Sugar Beet Cell Walls1

Solid-state nuclear magnetic resonance relaxation experiments were used to study the rigidity and spatial proximity of polymers in sugar beet (Beta vulgaris) cell walls. Proton T1ρ decay and cross-polarization patterns were consistent with the presence of rigid, crystalline cellulose microfibrils with a diameter of approximately 3 nm, mobile pectic galacturonans, and highly mobile arabinans. A direct-polarization, magic-angle-spinning spectrum recorded under conditions adapted to mobile polymers showed only the arabinans, which had a conformation similar to that of beet arabinans in solution. These cell walls contained very small amounts of hemicellulosic polymers such as xyloglucan, xylan, and mannan, and no arabinan or galacturonan fraction closely associated with cellulose microfibrils, as would be expected of hemicelluloses. Cellulose microfibrils in the beet cell walls were stable in the absence of any polysaccharide coating.