On a class of operators


Bogoljub Stankovi'c




In the field of Mikusiński operators [2] with defined convergence classes ([2], p.144) the series of the form (1), where $\lambda$ is a complex number and s differential operator, play an important role. Such a series defines an exponential operator $e^{-\lambda s^\alpha}$ which occurs when we apply Mikusiński operators in solving partial differential equations for numerical functions ([2] pp.439--453). In the mathematical literature the problem of convergence of such a series and the conditions when the series (1) represents an element from $\mathcal C$ are treated separately. This paper presents a very short proof which answers both questions. n PROPOSITION. \emph{The series $(1)$ converges for all $\lambda$ complex. If $|\arg\lambda|<\frac\pi2(1-\alpha)$, the series $(1)$ represents the function $t^{-1}\phi(0,-\alpha;-\lambda t^{-\alpha})$ which is an element of $\mathcal C$.} \emph{$\phi(\beta,-\alpha;z)$ is the function analysed by E.\,M. Wright} [6].