Some extensions of the Skolem-Mahler-Lech theorem (Q909695)

According to the Skolem-Mahler-Lech theorem, if the Taylor expansion of a rational function has infinitely many zero coefficients then the set of indices of those zero coefficients forms a finite set of arithmetic progressions modulo a finite set - that is, the zero coefficients occur periodically (eventually). The author discusses possible generalizations to Taylor expansions of wider classes of functions. He notes that periodicity of the Taylor expansion certainly does not hold for as wide a class as that of functions satisfying an algebraic differential equation and gives other classes for which the property does not hold. On the other hand he shows that the p-adic method can be stretched to a wider class of functions than just the exponential polynomials relevant to the case of rational functions. It remains open whether all power series satisfying linear differential equations with polynomial coefficients have the property that their zero coefficients occur periodically.