Uniform convergence of rows of the Padé table for functions with smooth Maclaurin series coefficients
Given a formal power series \(f(z):=\sum^{\infty}_{j=0}a_ jz^ j\) for which the quantity \(a_{j-1}a_{j+1}/a^ 2_ j\) has a prescribed asymptotic behavior as \(j\to \infty\), we obtain the asymptotic behavior of poles of rows of the Padé table, and the associated Toeplitz determinants. In particular, we can show for large classes of entire functions of zero, finite, and infinite order (including the Mittag- Leffler functions) and for \(N=1,2,3,...\), that the poles of [m/n](z) approach \(\infty\) with rate \(a_ m/a_{m+1}\) as \(m\to \infty\).
scientific article; zbMATH DE number 4006895
scientific article