Strengthening the Convergence of a Continued Fraction

V. E. Kornilov

(Presented by the Department of Higher Mathematics)

In this article an example is given of the asymptotic convergence of a continued fraction to a certain function in the domain of the complex variable \(w\) specified below in the text

\[ \varphi(w)=P_{2\kappa}(w)\cdot[Q_{2\kappa}(w)]^{-1}+R_{2\kappa}(w),\ |w|>1. \tag{1} \]

The function \(\varphi(w)\) depends on the function \(f(z)\) and is chosen in such a way that, under a subsequent transformation of suitable fractions \(P_k(w)[Q_k(w)]^{-1}\), it is possible to obtain a certain approximation for the function \(f(z)\):

\[ f(z)=P_{2\kappa}(z)[q_{2\kappa}(z)]^{-1}+r_{2\kappa}(z),\ |z|>1. \tag{2} \]

The method set forth for strengthening the convergence of a continued fraction can also be successfully applied to other continued fractions with positive link terms.

1. Let us multiply the terms of series (3) by the factor \(\sqrt{(-1)^\kappa}=i^\kappa\) and separate the real part of the series, etc.; after \(m\)-fold identical transformations we obtain the following series (4):

\[ \varphi(w)=\sum_{\kappa=0}^{\infty} p^{\kappa^2}\left(-\frac{1}{w}\right)^n;\quad p>1,\ |w|\gg1; \tag{3} \]

\[ f(z)=\sum_{\kappa=0}^{\infty} q^{\kappa^2}\left(-\frac{1}{z}\right)^\kappa;\quad p=q^{4^{-m}},\ |w|=|z|^{2^{-m}}, \tag{4} \]

With respect to series (3), the continued fraction (the determinants \(A_n, B_n\) are computed ([1], p. 28), and then \(a_j, b_j\)):

\[ \varphi(w)= \cfrac{1}{1} +\cfrac{p}{w} +\ldots+ \cfrac{p^{2n-1}(p^{2n}-1)}{1} +\cfrac{p^{4n+1}}{w} +\ldots \tag{5} \]

has the following sum of the terms \(a_j\) with odd indices:

\[ \sum_{n=0}^{\infty} a_{2n+1} = 1+\sum_{1=n}^{\infty} \frac{p^{n^2-n}}{(p^2-1)\ldots(p^{2n}-1)} > e^{\frac{n}{p^2-1}} = e^h. \tag{6} \]

On the basis of equalities (4) and (6) we obtain

\[ q=e^\sigma,\quad m\simeq(\ln 2\sigma h)(\ln 8)^{-1} \]

and, for a given \(m\) or \(h\), we can find \(h\) or \(m\).

1. The continued fraction (5), with respect to the function \(\varphi(w)\), converges asymptotically in the plane of the complex variable \(w\), except for the disk \(R \leqslant 1\) and the cut \([R,-\infty]\). The asymptotic convergence follows from the fact that

1) For any \(\rho>1\) there exists an \(m\) such that \(\rho^2>1+\rho^{2-2m}\); then

\[ \sum_{n=m}^{\infty} \frac{\rho^{n^2-n}}{(\rho^2-1)\ldots(\rho^{2n}-1)} = \sum_{k=0}^{\infty} \frac{C\rho^{k(k+2m-1)}}{(\rho^{2m}-1)\ldots(\rho^{2m+2k}-1)} < \]

\[ < \frac{C(C\rho^{2m}-1)}{\rho^{2m}-1-\rho^{2m-2}}, \]

and the continued fraction converges to at least two different functions ([1], p. 5).

2) The convergents \(P_k:Q_k\) converge asymptotically to the function \(\varphi(w)\) in the above-named domain by virtue of the following equality ([2], p. 466):

\[ \lim_{w\to\infty} w^{2n-1}\{\varphi(w)-P_{2n}:Q_{2n}\}=0. \tag{7} \]

Let us prove equality (7). As \(w\to\infty\), the continued fraction (5) converges to the function \(\varphi(w)\) by virtue of \(\lim_{w\to\infty} w\sum_{n=1}^{\infty} a_{2n}=\infty\). For any \(w\) in the domain \(\|w\|>1\), \(\arg w\ne\pi\), the exact value of the upper bound \(\sup |P_k:Q_k-P_{2n}:Q_{2n}|\) will in the general case be attained for \(k=2n+2m+1\) \((m=0,1,\ldots)\); then ([3], p. 450)

\[ \lim_{w\to\infty} \left|w^{2n-1}\{\varphi(w)-P_{2n}:Q_{2n}\}\right| \leqslant \]

\[ \leqslant \lim_{w\to\infty} \left\{ w^{2n-1}\left|\sup |P_k:Q_k-P_{2n}:Q_{2n}|\right| \right\} = \]

\[ = \lim_{w\to\infty} \left| \frac{ w^{2n-1}\rho^{3n^2-n}(\rho^2-1)\ldots(\rho^{2n}-1)(w^m+\ldots) }{ (w^{2n+m}+\ldots) } \right| =0, \]

and equality (7) is proved.

1. On the basis of the fact that the numerical value of the entire sum (6) and of its partial sum can be very large, we can (by bringing \(\rho\) sufficiently close to unity) obtain approximations for \(\varphi(w)\) such that for \(w=1\) and for some \(k\), \(|R_{2k}|<\varepsilon\), where, let us suppose, \(\varepsilon\simeq 10^{-10}\). Further, \(w\) in the convergent (1) is multiplied by \(i\), and its real part is separated, etc.; that is, we successively transform the convergent (1) \(m\) times and, replacing in the resulting approximation \(w\) by the variable \(z\), obtain an approximation of the form (2) for the function \(f(z)\). In this way we obtain an approximation for \(f(z)\) in the domain \(|z|>1\), \(\arg z\ne\pi\), such that the remainder term \(|r_{2k}|<\delta\), where, let us suppose, \(\delta\simeq 10^{-5}\). On the basis of the foregoing, \(\varepsilon\) depends on \(\delta,m,k\): \(\varepsilon=\psi(\delta,m,k)\).

REFERENCES

1. T. I. Stieltjes. Investigations on continued fractions. Moscow, ONTI, 1936.

2. M. A. Lavrent’ev and B. V. Shabat. Methods of the Theory of Functions of a Complex Variable. Moscow, Fizmatgiz, 1965.

3. S. S. Khloponin. Scientific Notes of the Mari Pedagogical Institute. Vol. XXVI, Yoshkar-Ola, 1965, pp. 445—486.