Padé tables of entire functions of very slow and smooth growth, II

Abstract

Letf(z):=Σ ^∞ _j=0 a _j z ^j, where a_j ≠ 0,j large enough, and for someq ε C such that ¦q¦<I,

$$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q,j \to \infty .$$

Define for m,n = 0,1,2,..., the Toeplitz determinant

$$D(m/n): = \det (a_{m - j + k} )_{j,k = 1}^n .$$

Given ɛ > 0, we show that form large enough, and for everyn = 1,2,3,...,

$$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$

We apply this to show that any sequence of Padé approximants {[m _k /n _k ]} ^∞₁ tof, withm _k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} ^∞₁ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n).