On singular points of meromorphic functions determined by continued fractions (Q722170)

Let a general continued-fraction be given, \[ \frac{a_1z^{\alpha_1}}{1+\frac{a_2z^{\alpha_2}}{1+\cdots}}=\mathcal K_{n=1}^\infty(\alpha_nz^{\alpha_n}/1) \] where \(a_n \in \mathbb C\backslash \{0\}\), \(\alpha_n \in \mathbb N\) for all \(n=1,2,\dots\) and \( \lim\limits_{n \to \infty} \alpha_n=\infty, \; \lim\limits_{n \to \infty} |a_n|^{1/\alpha_n}=1\). Leighton's Conjecture. The continued-fraction converges uniformly in the spherical metric on compact subsets of the disk \(\mathbb D=\{|z|<1\}\) to a meromorphic (in \(\mathbb D\)) function \(f\), and the disk \(\mathbb D\) is the natural domain of existence of the meromorphic function \(f\). In [Sb. Math. 197, No. 10, 1405--1416 (2006; Zbl 1151.30022); translation from Mat. Sb. 197, No. 10, 3--14 (2006)], \textit{A. A. Gonchar} proved a theorem, which implies, in particular, that Leighton conjecture holds for a nondecreasing sequence of exponents without any additional assumptions. In this paper it is shown that the Leighton conjecture holds also for meromorphic functions represented by continued fractions \(\mathcal K_{n=1}^{\infty}(a_n A_n(z)/1)\), where \(A_1, A_2,\dots\) is a sequence of polynomials with limit distribution of zeros whose degrees tend to infinity.