Lindelöf Representations and (Non-)Holonomic Sequences

Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but largely forgotten chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotics of certain finite difference sequences come out as a byproduct of our approach. Introduction There has been recently a surge of interest in methods for proving that certain sequences coming from analysis or combinatorics are non-holonomic. Recall that a sequence (fn) is holonomic, or P -recursive if it satisfies a linear recurrence with coefficients that are polynomial (equivalently, rational) in the index n; that is,