Meromorphic order from Taylor coefficients, least singular values of Hankel matrices and rational approximants

We introduce a natural notion of generalized order for entire and meromorphic functions and express it in terms of their Taylor coefficients. More precisely, one formula uses the least singular values of Hankel matrices of the coefficients, and the other uses their determinants. These formulas generalize the classical coefficient formula for the order of entire functions and its extensions by Edrei and Sheremeta. The proof combines a theorem of Pommerenke with a bound for the Nevanlinna characteristic that results from a convergence theorem for rational approximants introduced by Prokhorov. We also characterize the largest class of growth scales for which the associated generalized order is well-behaved.