Abstract. Suppose that $f (z) = \sum ^\infty_{\nu = 0} a_\nu z^\nu $ is a power series with radius of convergence 1 and let $A = [\alpha_{n \nu }]$ be a lower triangular matrix. We are interested in the behaviour of the $A$-transforms $$ \sigma_n (z) = \sum ^n_{\nu = 0} \alpha_{n \nu } \sum ^{\nu }_{\mu = 0} a_\mu z^\mu, $$ where we do not assume that the matrix $A$ is regular or satisfies some quasi-regular properties. Among others we deal with the following problems. \item{$\bullet $} Let it be given an $R > 0$. Under which necessary and sufficient conditions on $A$ is $\{\sigma_n (z) \} $ compactly convergent in $|z| < R$ and which limit functions are possible. \item{$\bullet $} The growth-properties of $\{\sigma_n (z)\} $ are investigated. \item{$\bullet $} Some problems concerning the value-distribution of the $A$-transforms are studied. For instance the location of the limit points of $w_0$-values of $\sigma_n (z)$, the number of $w_0$-values in a circle $|z| < S$ and properties of the mapping $w = \sigma_n (z)$ are investigated.

AMS Subject Classification (1991): 40A25, 40G99

Received April 15, 1999. (Registered under 2723/2009.)