$J$-symmetrical functions and series in the complex plane

In the recent paper [1,2,3] the authors have introduced the notion of $(j,k)$-symmetrical functions, proved several properties of these functions and given their different applications. In the present paper the authors extend the considerations onto the $j$-symmetrical functions. They deduce the general form of $j$-symmetrical functions (thm.1) and show some criteria of expandability of a function into series with $j$-symmetrical components (thm.3 and thm.4).