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.

Functional implications for the microtubule-associated protein tau: localization in oligodendrocytes.

We present evidence that the microtubule-associated protein tau is present in oligodendrocytes (OLGs), the central nervous system cells that make myelin. By showing that tau is distributed in a pattern similar to that of myelin basic protein, our results suggest a possible involvement of tau in some aspect of myelination. Tau protein has been identified in OLGs in situ and in vitro. In interfascicular OLGs, tau localization, revealed by monoclonal antibody Tau-5, was confined to the cell somata. However, in cultured ovine OLGs with an exuberant network of processes, tau was detected in cell somata, cellular processes, and membrane expansions at the tips of these processes. Moreover, in such cultures, tau appeared localized adjacent to or coincident with myelin basic protein in membrane expansions along and at the ends of the cellular processes. The presence of tau mRNA was documented using fluorescence in situ hybridization. The distribution of the tau mRNA was similar to that of the tau protein. Western blot analysis of cultured OLGs showed the presence of many tau isoforms. Together, these results demonstrate that tau is a genuine oligodendrocyte protein and pave the way for determining its functional role in these cells.