A generalization of the Jacobi's triple product formula and some applications [Une généralisation de la formule du triple produit de Jacobi et quelques applications A generalization of the Jacobi's triple product formula and some applications]

摘要

If An≠0 for all n∈Z, we show the series with 2 variables Q(x,y)=σ_n∈ZAnxny~(n(n+1)/2) factorizes formally in an infinite triple product, which tgeneralizeshe Jacobi's formula. Let ρo be the positive root of σ_(k=1)ρΩ~(k2)=1/2, we prove the convergence of the factorization of Q for x∈C* and |y|<Ωo2ρΩ1 with Ω=sup_n∈Z|A_(n-1An+1)/A_n~2|. We deduce that if Ω<ρo2=0.2078. each zero of the Laurent series f(x)=σ_n∈ZA_nx~n can be explicitly calculated as the sum or the inverse of the sum of series, whose terms are polynomial expressions of A_(n-1An+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 σ(n) of an integer.