Complements to theorems of M. Blambert - R. Parvatham about singularities of an analytic function defined by an \(E\)-Dirichletian element (Q5929784)

Let \(\{f\}:\sum^{+\infty}_{n=1}f_n(s)\exp(-\lambda_ns)\), where \(f_n(s)=P_n(s) \exp(\alpha_ns)\) are entire functions of bounded indices \(\nu_n\), \(P_n(s)\) are polynomials of degree \(m_n(>0)\), \(\{\alpha_n\}\) and \(\{\lambda_n\} \subset\mathbb{C}\) and \(0<|\lambda_n|\uparrow+ \infty\). The following results are proved: 1) If \(D^*=\lim\sup \{\sum^n_{j=1} (m_j+1)/ \lambda_n\} <+\infty\), \(m_n\sim\nu_n\) and \(|\lambda_n |\sim |\lambda_n -\alpha_n |\), then \(\{f\}\) converges absolutely and locally uniformly on \({\mathcal D}_{*,\beta^*}\) and diverges on \((\mathbb{C}\setminus {\mathcal E}^*) \setminus\overline {\mathcal D}_{*,0}\), where \({\mathcal D}_{*,\alpha} =\{s\in \mathbb{C} \setminus {\mathcal E}^*: \delta_*(s) >\alpha\}\), \({\mathcal E}^*= {\mathcal E}^d\cup {\mathcal E}_\infty\), \({\mathcal E}^d\) being the derived set of \({\mathcal E}\), the set of all zeroes of \(P_n(n=1,2, \dots)\) and \({\mathcal E}_\infty\) consisting of points which are zeroes of infinitely many \(P_n\), \(\beta^* =\lim\sup \{m_n/|\lambda_n |\}\) and \(\delta_*(s)= \lim\sup \{-\log|P_n(s)\exp (-s(\lambda_n-\alpha_n))/ |\lambda_n|\}\). Consider an entire function \(g\) of the finite type \(T\) of order 1 which has zeroes \(\lambda_n\) of order \(m_n+1\) \((n=1, 2, \dots)\). Let denote \({\mathcal C}\) the image of the indicator diagram with respect to the imaginary axis and \(\gamma^*\) a number deduced from \(g\). 2) Under the conditions in 1) and \(\gamma^*< +\infty\), \(\{f\}\) has no holomorphic extension by means of \({\mathcal C}\) from \(\{s\in \mathbb{C}:s= s_1+s',s' \in{\mathcal C}\} \subset {\mathcal D}_{*, \beta}*\) to \(\{s\in \mathbb{C}:s= s_2+s',s' \in{\mathcal C}\}\), where \(s_2 \in (\mathbb{C}\smallsetminus {\mathcal E}^{t-1)_*}) \smallsetminus \overline{\mathcal D}_{*,0}^{{\mathcal E}(t-1)}(t=1,2, \dots, T_0+1)\), \({\mathcal E}^{(t-1)}\) etc. being defined by \(\{P_n^{(t-1)}\}\) as \({\mathcal E}\) etc. by \(\{P_n\}\).