AbstractVarious sequences that possess explicit analytic expressions can be analysedasymptotically through integral representations due to Lindel¨of, which belong to anattractive but somewhat neglected chapter of complex analysis. One of the outcomesof such analyses concerns the non-existence of linear recurrences with polynomialcoefficients annihilating these sequences, and, accordingly, the non-existence of lin-ear differential equations with polynomial coefficients annihilating their generatingfunctions. In particular, the corresponding generating functions are transcendental.Asymptotic estimates of certain finite difference sequences come out as a byproductof the Lindel¨of approach...