## On the Ritt order and type of a certain class of functions defined by $$BE$$-Dirichletian elements.(English)Zbl 0963.30003

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.

### MSC:

 30B50 Dirichlet series, exponential series and other series in one complex variable

### Keywords:

Ritt order; $$BE$$-Dirichletian elements
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