On some non-holonomic sequences

Abstract A sequence of complex numbers is holonomic if it satisfies a linear recurrence with polynomial coefficients. A power series is holonomic if it satisfies a linear differential equation with polynomial coefficients, which is equivalent to its coefficient sequence being holonomic. It is well known that all algebraic power series are holonomic. We show that the analogous statement for sequences is false by proving that the sequence f p ngn is not holonomic. In addition, we show that fn n gn, the Lambert W function and flog ngn are not holonomic, where in the case of flog ngn we have to rely on an open conjecture from transcendental number theory