Analytic Continuation of Taylor-Dirichlet Series and Non-Vanishing Solutions of a Differential Equation of Infinite Order

摘要

In the spirit of the Fabry-Polya Gap Theorems on the singularities of power series, we investigate similar phenomena for Taylor-Dirichlet seriesrn∞∑(μ_n-1 ∑ j=0 c_(n,j)z~j)e~(λ_nz)rnassociated to the positive real multiplicity-sequence ∧ = {λ_n,μ_n}∞n=1. Assume that ∧ has positive finite density d, counting multiplicities, and that ∧ belongs to a certain class that we denote by U(d,0). LetrnF(z)=∞∏n=1(1-z~2/λ_n~2)~μ_n and g(z,w)=exp(wz)/F(w).rnThen the seriesrn∞ ∑ n=1(μ_n-1 ∑ j=0 c_(n,j)z~j)e~λ_n~z = ∑ res g(z,λ_n), defines an analytic function in the open left half-plane C_. This function cannot be extended analytically across any open interval lying on the imaginary axis having length greater than 2nd. Nevertheless, this series can be extended analytically as an even function to the open right half-plane C_+ across the open segment (-iπd, iπd). Applications are given to a differential equation of infinite order. Let D = d/dx and L(z) = zF{z). The equation L(D)f{x) = 0 with the boundary condition lim_(x→±∞) f(x) = 0 has non-vanishing solutions. Extensions are given with D = d/dz.