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.

Human milk lactoferrin inactivates two putative colonization factors expressed by Haemophilus influenzae

Haemophilus influenzae is a major cause of otitis media and other respiratory tract disease in children. The pathogenesis of disease begins with colonization of the upper respiratory mucosa, a process that involves evasion of local immune mechanisms and adherence to epithelial cells. Several studies have demonstrated that human milk is protective against H. influenzae colonization and disease. In the present study, we examined the effect of human milk on the H. influenzae IgA1 protease and Hap adhesin, two autotransported proteins that are presumed to facilitate colonization. Our results demonstrated that human milk lactoferrin efficiently extracted the IgA1 protease preprotein from the bacterial outer membrane. In addition, lactoferrin specifically degraded the Hap adhesin and abolished Hap-mediated adherence. Extraction of IgA1 protease and degradation of Hap were localized to the N-lobe of the bilobed lactoferrin molecule and were inhibited by serine protease inhibitors, suggesting that the lactoferrin N-lobe may contain serine protease activity. Additional experiments revealed no effect of lactoferrin on the H. influenzae P2, P5, and P6 outer-membrane proteins, which are distinguished from IgA1 protease and Hap by the lack of an N-terminal passenger domain or an extracellular linker region. These results suggest that human milk lactoferrin may attenuate the pathogenic potential of H. influenzae by selectively inactivating IgA1 protease and Hap, thereby interfering with colonization. Future studies should examine the therapeutic potential of lactoferrin, perhaps as a supplement in infant formulas.