Hadamard Composition of Series in Systems of Functions

Abstract

For regularly converging in ${\Bbb C}$ series $A_j(z)=\sum\limits_{n=1}^{\infty}a_{n,j}f(\lambda_nz)$, $1\le j\le p$, where $f$ is an entire transcendental function, the asymptotic behavior of a Hadamard composition $A(z)=\break=(A_1*...*A_p)_m(z)=\sum\limits_{n=1}^{\infty} \left(\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}a_{n,1}^{k_1}\cdot...\cdot a_{n,p}^{k_p}\right)f(\lambda_nz)$ of genus m is investigated. The function $A_1$ is called dominant, if $|c_{m0...0}||a_{n,1}|^m \not=0$ and $|a_{n,j}|=o(|a_{n,1}|)$ as $n\to\infty$ for $2\le j\le p$. The generalized order of a function $A_j$ is called the quantity $\varrho_{\alpha,\beta}[A_j]=\break=\varlimsup\limits_{r\to+\infty}\dfrac{\alpha(\ln\,\mathfrak{M}(r,A_j))}{\beta(\ln\,r)}$, where $\mathfrak{M}(r,A_j)=\sum\limits_{n=1}^{\infty} |a_{n,j}|M_f(r\lambda_n)$, $ M_f(r)=\max\{|f(z)|:\,|z|=r\}$ and the functions $\alpha$ and $\beta$ are positive, continuous and increasing to $+\infty$. Under certain conditions on $\alpha$, $\beta$, $M_f(r)$ and $(\lambda_n)$, it is proved that if among the functions $A_j$ there exists a dominant one, then $\varrho_{\alpha,\beta}[A]=\max\{\varrho_{\alpha,\beta}[A_j]:\,1\le j\le p\}$. In terms of generalized orders, a connection is established between the growth of the maximal terms of power expansions of the functions $(A^{(k)}_1*...*A^{(k)}_p)_m$ and $((A_1*...*A_p)_m)^{(k)}$. Unresolved problems are formulated