On the growth of meromorphic functions (Q5899713)

It is well known, and found in any second course in analytic function theory, that the order and type of an entire function f can be determined from the limiting behaviour of the Taylor coefficients of f. It is almost as well known that the growth of f can also be determined from the degree of best approximation to f by polynomials on (say) the unit disc. If g is f/q where q is a polynomial with m zeros the growth of g is that of f and the above theory for f applies. The author has previously [Zesz. Nauk Uniw. Jagielloń 661, Pr. Mat. 24, 177-187 (1984; Zbl 0551.30033)] proved a similar result for looking at coefficients of (n/m) Padé approximants rather than (Taylor pol's)/q. He has generalized the result about degree of approximation to sequences of rationals of type \((n/m_ n)\) where \(m_ n\) tends slowly to infinity so it is not necessary to know m for application of the result [see Analysis 8, 397-406 (1988; Zbl 0664.30036)]. This paper contains the result for coefficients of the \((n/m_ n)\) Padé approximants.