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Let E be a complex Banach space and U be an open subset of E. The space of all holomorphic functions on U will be represented by H(U). We prove the following results: Theorem 1. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is lineable. This means that E(U), together with 0, contains an infinite dimensional vector subspace. Theorem 2. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is c-lineable. This means that E(U), together with 0, contains a vector subspace of dimension c, the cardinality of the continuum. (Of course Theorem 1 follows from Theorem 2, but the proof of Theorem 1 is presented in a much simpler way.) Theorem 3. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is algebrable. This means that E(U), together with 0, contains a subalgebra which is generated by an infinite algebraically independent set.

Let $X$ be a Riemann domain over $\mathbb C^k\times\mathbb C^\ell$. If $X$ is domain of holomorphy with respect to a family $\mathcal F\subset\mathcal O(X)$, then there exists a pluripolar set $P\subset\mathcal C^k$ such that every slice $X_a$ of $X$ with $a\notin P$ is a domain of holomorphy with respect to the family $\{f|_{X_a}: f\in\mathcal F\}$., Comment: 7 pages

We give a characterization of $L_h^2$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively., 9 pages

Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $\gamma_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that the $2n$th root of $\gamma_{2n}$ has a limit, called the cogrowth exponent with respect to generators $g_1,g_2,..., g_r.$ We show by analytic methods that the numbers $\gamma_n$ vary regularly; i.e. the ratio $\gamma_{2n+2}/\gamma_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $\gamma(z),$ the generating function associated with the coefficients $\gamma_n.$