Padé approximation and its applications location:Bad Honnef date:7-10 March 1983
If a rational function is analytic at the origin one can consider its Taylor series at the origin and construct the qd-table associated with it. It is well known that the limits of certain columns in this table will give information about the poles of the function and the limits of certain rows will give information about the zeros. What if one knows the Laurent series of that function in an annular region around the origin? With this double infinite series we can associate a double infinite qd-table. It is known that the downward limits of certain columns will give again some of the poles of the function (those around infinity) and the upward limits will give the other poles (those around the origin). It turns out that the rows of this table give the zeros of the function as limits.
An algorithm essentially equivalent to the πζ or uv algorithms and analogous to the qd and FG algorithms is developed and the limit of the rows gives information about the zeros of the function in question which has an infinite principal part in its Laurent expansion. The main tool of the proofs is a theorem of K. M. Day [Trans. Amer. Math. Soc. 206 (1975), 224-245; MR0379803 (52 #708)] on the asymptotic behavior of Toeplitz determinants.