In the spirit of the Fabry-Polya Gap Theorems on the singularities of power series, we investigate similar phenomena for Taylor-Dirichlet series $$\sum\limits_{n = 1}^\infty {\left( {\sum\limits_{j = 0}^{\mu _n - 1} {c_n j^{z^j } } } \right)e^{\lambda _n z} }$$ associated to the positive real multiplicity-sequence Λ = {λn, μn}n=1∞. Assume that Λ has positive finite density d, counting multiplicities, and that Λ belongs to a certain class that we denote by U(d, 0). Let $$F(z) = \prod\limits_{n = 1}^\infty {\left( {1 - \frac{{z^2 }} {{\lambda _n ^2 }}} \right)^{\mu _n } } and g\left( {z,w} \right) = \frac{{\exp (wz)}} {{F(w)}}.$$ Then the series $$\sum\limits_{n = 1}^\infty {\left( {\sum\limits_{j = 0}^{\mu _n - 1} {c_{n,j} z^j } } \right)e^{\lambda _n z} = \sum {\operatorname{Re} s g} } \left( {z,\lambda _n } \right),$$ 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 2πd. 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 limx→+ -∞ f(x) = 0 has non-vanishing solutions. Extensions are given with D = d/dz.