Power series, rational functions, and certain determinants

We consider a fundion f of a complex variahl(' and ('Xamiuc a IH'C('Ssary and sufficient. condition for .f( 1/ z) t.o lH' rational. Tlw prohlPm of ddcnuininp; such conditions has arisen while considerin1!; tmiqneness sets for holo111orphic fund.ious in several variables (Harris (5].) It also appears in control theory <liHl in the work of Byrnes (1]. As may be expected, such a condition uwy lH' CXJ>I·c·sscd usiug the coefficients in the power scrics for f. More specifically, .f(1/z) is rational if <llld only if then' is a recurrence relation of a particular form amo11p; t.lw codficic•nt s. \Vf' prcsent. the general solution of such recurrence rcbdions and usc· it. t.o arrin· at. a closed form for a power series whose coefficicuts s;disfy such il H'CillTC'Il!"f' relation. \Ve also develop a simple expression for a certain class of dd.fTillinants of which a Vandermonde determinant is a special c;1sc. Besides being used in the proof of the general solution of a recurn~nce relation, this expression has applications elsewhere, including the interpolation work of 1vlmt iu, :tvlilkr, nml Pearce (G].