9 problems

Let $A$ be a finite subset of $\mathbb{C}$, and let $f\in\mathcal{A}(\overline{\mathbb{C}}\setminus A)$ be a germ analytically continuable along every path in…

Let $f$ be an algebraic function analytic at $\infty$, and let $[n/n]$ denote its diagonal Padé approximants. A spurious pole is a pole of a Padé approximant in the sense defined i…

Consider the $[P/P]$-Padé approximant $R_P(z)=T_P(z)/Q_P(z)$, where $T_P$ and $Q_P$ have degree $P$ and are computed from the first $2P+1$ Taylor coefficients of $f$. Let $t$ denot…

Coefficient matching conjecture. In the limit as $m$ tends to infinity, the continued fraction $g_m(n)$ agrees with $B(n)$; equivalently, their numerator sequences satisfy

Let $f_0$ and $f_1$ be the functions associated with disjoint admissible compact sets $F_0\in\mathcal F_0$ and $F_1\in\mathcal F_1$, and let $\Gamma=(\Gamma_0,\Gamma_1)$ be a vecto…

Let $F\in\mathcal F_f$ be an admissible compact set for $f$, and let $\{\Omega_n\}$ be an interpolation table disjoint from $F$. Assume that the limiting density $\omega$ in the in…

Let $f(z)=\sum_{i=0}^\infty f_i z^{-i}$ be a convergent power series, and let $[n/n]_f(z)=p_n(z)/q_n(z)$ be its $n$-th diagonal Padé approximant in reduced form, with $q_n$ monic o…

Strong asymptotics conjecture.

Let ${\bf f}=(f_1,\ldots,f_d)$ and ${\bf m}=(m_1,\ldots,m_d)\in\mathbb{Z}_+^d\setminus\{{\bf 0}\}$, and suppose that the denominator $Q_{n,\bf m}$ is unique for all sufficiently la…