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.

Results of MDR-1 vector modification trial indicate that granulocyte/macrophage colony-forming unit cells do not contribute to posttransplant hematopoietic recovery following intensive systemic therapy

To formally test the hypothesis that the granulocyte/macrophage colony-forming unit (GM-CFU) cells can contribute to early hematopoietic reconstitution immediately after transplant, the frequency of genetically modified GM-CFU after retroviral vector transduction was measured by a quantitative in situ polymerase chain reaction (PCR), which is specific for the multidrug resistance-1 (MDR-1) vector, and by a quantitative GM-CFU methylcellulose plating assay. The results of this analysis showed no difference between the transduction frequency in the products of two different transduction protocols: “suspension transduction” and “stromal growth factor transduction.” However, when an analysis of the frequency of cells positive for the retroviral MDR-1 vector posttransplantation was carried out, 0 of 10 patients transplanted with cells transduced by the suspension method were positive for the vector MDR-1 posttransplant, whereas 5 of 8 patients transplanted with the cells transduced by the stromal growth factor method were positive for the MDR-1 vector transcription unit by in situ or in solution PCR assay (a difference that is significant at the P = 0.0065 level by the Fisher exact test). These data suggest that only very small subsets of the GM-CFU fraction of myeloid cells, if any, contribute to the repopulation of the hematopoietic tissues that occurs following intensive systemic therapy and transplantation of autologous hematopoietic cells.