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Content available remote Zofia Szmydt (1923-2010)
EN
We derive differential relations between the Dunkl spherical and solid means of continuous functions. Next we use the relations to give inductive proofs of mean-value properties for the Dunkl polyharmonic functions and their converses.
3
Content available remote Singular linear differential equations and Laurent series
EN
Linear ordinary differential operators with meromorphic coefficients at zero are studied. It is well known that in the case when zero is a regular or regular singular point then fundamental system of solutions consists of convergent series of the Taylor type. On the other hand in the case of irregular singular point power series solution, in general, does not converge; however it can be asymptotically sum up in sectors to an exact solution. The aim of the paper is to show that for a class of operators with irregular singular point the fundamental system of solutions can be found in a form of convergent Laurent type series of a Gevrey order. Under suitable conditions the convergence of the approximation scheme for a functional equation Wj ( z -j) G ( z - j) = H ( z 't j is also j=-k derived and properties of its solution G are described.
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