Power series equivalent to rational functions: A shifting-origin kronecker type theorem, and normality of Padé tables
Tóm tắt
Letf(z) be a function analytic in a neighbourhood of zero. For each pair of non-negative integers (m, n), form then byn Toeplitz determinantD(m/n) whose entries are the Maclaurin series coefficients off, namely,
$$D(m/n): = det[f^{(m + j - k)} (0)/(m + j - k)!]_{j,k = 1'}^n $$
where we definef
(s)
(0)/s!≔0, ifs<0. A classical theorem of Kronecker asserts thatf(z) is a rational function if and only if there existm
0 andn
0 such thatD(m/n)=0 form≧m
0 andn≧n
0. In some important recent work, such as the solution of Meinardus's Conjecture, it has been found useful to form Padé approximants not at 0, but at different points near 0. In questions regarding normality of these Padé approximants with a shifting origin, one considers then byn determinantD(m/n; u) which is defined by (1), but with 0 replaced byu. In this spirit, we prove thatf(z) is a rational function if and only if there exists asingle pair of positive integers (m, n) such thatD(m/n; u) is identically zero foru in a neighbourhood of zero. Further, we deduce that except possibly for countably many values ofu, the Padé table of a non-rationalf(z) atz=u is normal, that isD(m/n; u)≠0, for allm, n=0, 1, 2,....
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