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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. (English) Zbl 0929.30004

Jones, William B. (ed.) et al., Orthogonal functions, moment theory, and continued fractions: theory and applications. Proceedings of a research conference/workshop, Campinas, São Paulo, Brazil, June 19–28, 1996. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 199, 257-274 (1998).
Let \(v(t)\) be a non–negative function on \((0,\infty)\) such that \[ c_k:=\int_0^\infty t^k v(t) dt;\qquad k=0,1,2,\dots \] exist for all \(k\). Then the Stieltjes transform \[ G(z):=\int_0^\infty {v(t)\over z+t} dt\quad\text{for} z\in S_\pi:=\{z\in\mathbb C: | \arg z| <\pi\} \] has an S–fraction expansion of the form \[ K:={b_1\over z} {\quad\atop +} {b_2\over 1} {\quad\atop +} {b_3\over z} {\quad\atop +} {b_4\over 1} {\quad\atop +} {b_5\over z} {\quad\atop +\dots};\quad b_n>0. \] The problem addressed in this paper is to find the asymptotic behavior of \(\{b_n\}\), which is important to determine and improve the convergence behavior of \(K\). By applying Freud’s conjecture proved by D. S. Lubinsky, H. N. Mhaskar and E. B. Saff [Constructive Approximation 4, 65-83 (1988; Zbl 0653.42024)], the authors prove that if \(v(t)=w_r(\sqrt{t})/\sqrt{t}\), where \(w_r(x)\) is an even weight function on \(\mathbb R\) of the form \[ w_r(x)=u(x)\psi(x)V(x)e^{-r| x| ^\alpha};\quad r>0, \alpha>0, \] where \(u(x)\) is a Jacobi factor, \(\psi\in L_\infty (\mathbb R)\) with \(\psi\geq 0\) and \(\lim_{x\to\pm\infty}\psi(x)=1\), and \(V>0\) satisfies \(\lim_{x\to\pm\infty}| x| ^{-\alpha}\text{Log} V(x)=0\) and some extra conditions, then \[ \lim_{n\to\infty}{b_n\over n^{2/\alpha}}={1\over 4} \left [ {\sqrt{\pi}\Gamma({\alpha\over 2})\over r \Gamma({\alpha+1\over 2})}\right ]^{2/\alpha}, \] and thus \(K\) converges to \(G(z)\) in \(S_\pi\) for \(\alpha\geq 1\).
By use of this result, the authors prove a conjecture by J. Cizek and E. R. Vrscay [C. R. Math. Acad. Sci., Soc. R. Can. 4, 201-206 (1982; Zbl 0493.33020)] on the Binet function \(J(z)\): Since \[ {J(\sqrt{z})\over\sqrt{z}}=\int_0^\infty{v(t)\over z+t} dt\quad \text{where} v(t):={1\over 2\pi\sqrt{t}} \text{Log}{1\over 1-e^{-2\pi\sqrt{t}}}; t>0, \] they find that the S–fraction expansion \(K\) of \(J(\sqrt{t})/\sqrt{t}\) satisfies \(\lim_{n\to\infty} (b_n/n^2)={1\over 16}\). In particular this means that the S–fraction converges to the “right value” in \(S_\pi\), and that its convergence is faster than the convergence of the power series expansion.
For the entire collection see [Zbl 0897.00031].

MSC:

30B70 Continued fractions; complex-analytic aspects
33B15 Gamma, beta and polygamma functions
40A15 Convergence and divergence of continued fractions
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