On some general classes of partial linear complex vector functional equations (Q2716858)

A typical generalized cyclic functional equation is written in this paper as NEWLINE\[NEWLINE \sum_{j=1}^k f_j(X_j,X_{j+1},\dots,X_{j+p-1},Y_j,Y_{j+1},\dots,Y_{j+q-1})=0, NEWLINE\]NEWLINE where \(X_{n+j}=X_j, Y_{n+j}=Y_j\) for all \(j\leq k\leq n.\) Several cases are investigated according to the relation of magnitude of \(p,q\) and \(n.\) For each case a theorem on the general solution is offered. The solution in each of them takes 6-10 lines of mathematical symbols, including 6-10 summation symbols.NEWLINENEWLINENEWLINEThe Zbl review of a paper on a similar subject by the present authors and by \textit{K. G. Trenčevski} [Appl. Sci. 2, No.~1, 13--18 (2000; Zbl 0954.39019); not quoted in the references of this paper] contained the sentences ``\dots the authors seem to be unaware of anything published since 1961, for example of \textit{D. Ž. Djoković}'s 1964 paper [Publ. Math. Inst. Hung. Acad. Sci., Ser. A 9, 51-59 (1964; Zbl 0199.47801)]''.NEWLINENEWLINENEWLINEThe present paper quotes (as reference [4]) that article, but still nothing later than 1967 (except a paper by the first author ``to appear'').NEWLINENEWLINENEWLINEIt is emphasized in this paper, as in the previous one by the present authors and by K. G. Trenčevski [loc. cit.], that the variables and the function values are in a finite dimensional complex vector space, but it seems (cf. the Djoković reference above) that the result and proof is unchanged when the variables are in an arbitrary set (having at least \(n\) elements) and the function values are in an \(n\)-divisible abelian group.