  Various sequences that possess explicit analytic expressions can be analysed
asymptotically through integral representations due to Lindel\"of, which belong
to an attractive but somewhat neglected 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. Asymptotic estimates of certain finite
difference sequences come out as a byproduct of the Lindel\"of approach.
