Une généralisation de la formule du triple produit de Jacobi et quelques applications

Abstract

If A n ≠ 0 for all n ∈ Z , we show the series with 2 variables Q ( x , y ) = ∑ n ∈ Z A n x n y n ( n + 1 ) / 2 factorizes formally in an infinite triple product, which generalizes the Jacobiʼs formula. Let ρ o be the positive root of ∑ k = 1 ∞ ρ k 2 = 1 / 2 , we prove the convergence of the factorization of Q for x ∈ C ⁎ and | y | < ρ o 2 Ω − 1 with Ω = sup n ∈ Z | A n − 1 A n + 1 / A n 2 | . We deduce that if Ω < ρ o 2 = 0.2078 … each zero of the Laurent series f ( x ) = ∑ n ∈ 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 σ ( n ) of an integer.