A generalization of the Jacobi's triple product formula and some applications

If A(n) not equal 0 for all n is an element of Z, we show the series with 2 variables Q (x, y) = Sigma(n is an element of Z) A(n)x(n)y(n(n+1)/2) factorizes formally in an infinite triple product, which generalizes the Jacobi's formula. Let rho(0) be the positive root of Sigma(infinity)(k=1) rho(k2) = 1/2, we prove the convergence of the factorization of Q for x is an element of C* and vertical bar y vertical bar < rho(2)(0)Omega(-1) with Omega = sup(n is an element of z) vertical bar A(n-1)A(n+1)/A(n)(2)vertical bar. We deduce that if Omega < rho(2)(0) = 0.2078 ... each zero of the Laurent series f(x) = Sigma(n is an element of Z) A(n)x(n) can be explicitly calculated as the sum or the inverse of the sum of series, whose terms are polynomial expressions of A(n-1)A(n+1)/A(n)(2). If the previous inequality is wide and f (x) real, then all its zeros are real numbers. An other application is when you know the triple product factorization of Q(x, y) by another way than described in the note, to identify them. So with the Jacobi theta function, we obtained a new identity for the sum of divisors sigma(n) of an integer. (C) 2011 Academie des sciences. Publie par Elsevier Masson SAS. bus droits reserves.