On the Ritt order and type of a certain class of functions defined by \(BE\)-Dirichletian elements (Q1587798)

Let \(0<\lambda_n\nearrow+\infty\) and let \(f_n(s)=P_n(s)\exp(\alpha_n s)\), where \(P_n(s)=\sum_{j=0}^{m_n}a_{n,j}s^j\) is a polynomial of degree \(m_n\) and \(|\alpha_n|\leq k<\lambda_1\) (\(k\) is independent of \(n\)). Consider two series: \(f_{\tau_0}(s):=\sum_{n=1}^\infty f_n(\sigma+i\tau_0)\exp(-s\lambda_n)\) (\(s=\sigma+i\tau\), \(\tau_0\in\mathbb R\)), \(f_A(s):=\sum_{n=1}^\infty A_n\exp(-s\lambda_n)\), where \(A_n:=\max\{|a_{n,j}|: j=0,1,\dots,m_n\}\). The author presents conditions under which the Ritt order and type of \(f_{\tau_0}\) and \(f_A\) coincide.