Explicit continued fractions and quantum gravity

This paper deals with the subject of completely integrable systems, particularly Painlevé equations, monodromy and Stokes parameters, complex analysis, approximation theory, computational mathematics, and number theory. The starting point is the rather narrow question: What is the closed-form expression for the continued fraction expansions of functions having closed (explicit) form definition?     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

Invariant measures for parabolic IFS with overlaps and random continued fractions

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no ``overlaps.' We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

     

Tasoev's continued fractions and Rogers-Ramanujan continued fractions

We show some new variations on Tasoev's continued fractions , where the periodic parts include the exponentials in k instead of the polynomials in k. We also mention some relations with other kinds of continued fractions, in particular, with Rogers-Ramanujan continued fractions.     

Quasi-periodic continued fractions

Using recent work of Adamczewski and Bugeaud, we are able to relax the conditions given by Baker to establish transcendence in the class of quasi-periodic continued fractions.     

Matrix-valued continued fractions

We discuss the properties of matrix-valued continued fractions based on Samelson inverse. We begin to establish a recurrence relation for the approximants of matrix-valued continued fractions. Using this recurrence relation, we obtain a formula for the difference between mth and nth approximants of matrix-valued continued fractions. Based on this formula, we give some necessary and sufficient conditions for the convergence of matrix-valued continued fractions, and at the same time, we give the estimate of the rate of convergence. This paper shows that some famous results in the scalar case can be generalized to the matrix case, even some of them are exact generalizations of the scalar results.     

Transcendental continued fractions

In two previous papers Nettler proved the transcendence of the continued fractions $\text{A := a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+ ?}$, $\text{B := b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+ ?}$ as well as the transcendence of the numbers A + B, A ? B, AB, $\text{A}\text{B}$ where the a's and b's are positive integers satisfying a certain mutual growth condition. In the present paper even the algebraic independence of A and B is proved under almost the same condition and furthermore a result concerning the transcendency of A^B is established.     

Transcendental continued fractions

In a previous paper it was proven that given the continued fractions

$\text{A = a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+… and B = b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+…}$

where the a's and b's are positive integers, then A, B, A ± B, $\text{A}\text{B}$ and AB are irrational numbers if $\text{a}{}_{\mathrm{n}}\text{2}\text{> b}{}_{\mathrm{n}}\text{> a}{}_{\mathrm{n?1}}{}^{\mathrm{5n}}$ for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, $\text{A}\text{B}$ and AB are transcendental numbers if a_n > b_n > a_n?1^(n?1)2 for all n sufficiently large.     

Relation between interpolating integral continued fractions and interpolating branched continued fractions

We construct and investigate an interpolating integral continued fraction, which represents a natural generalization of interpolating continued fractions. We also point out the optimal choice of the sequence of interpolating knots.     

An algorithm for calculating continued fractions

A new algorithm is described in the paper for calculating continued fractions. The conditions are given under which this algorithm is faster than the hitherto fastest algorithm for handling this problem. Also the interrelation between the suggested algorithm and that for calculating a tridiagonal system of linear equations is investigated.     

Ergodic theory for complex continued fractions

For a complex continued fraction algorithm the invariant measure for the shift transformation is determined explicitly in terms of elementary functions, and the transformation is shown to be ergodic. Analogues of well-known theorems of Khintchine and Lévy are obtained.     

Pringsheim's theorem for generalized continued fractions

In this paper we present a generalization to generalized continued fractions of Pringsheim's theorem on the convergence of ordinary continued fractions.     

Hausdorff measures and packing measures of limit sets of CIFSs of generalized complex continued fractions

ABSTRACT

We consider a certain family of CIFSs of the generalized complex continued fractions with a complex parameter space. We show that for each CIFS of the family, the Hausdorff measure of the limit set of the CIFS with respect to the Hausdorff dimension is zero and the packing measure of the limit set of the CIFS with respect to the Hausdorff dimension is positive (main result). This is a new phenomenon of infinite CIFSs which cannot hold in finite CIFSs. We prove the main result by showing some estimates for the unique conformal measure of each CIFS of the family and by using some geometric observations.     

An algorithm for calculating generalized continued fractions

In this paper the generalization of a continued fraction in the sense of the Jacobi-Perron algorithm (called an n-fraction) is considered.Apart from the known algorithms to calculate an n-fraction a new one is derived and the algorithms are compared with respect to the number of operations required and the time to execute these operations.     

The Euler-Minding series for branched continued fractions

In the paper “Branched continued fractions for double power series” [J. Comput. Appl. Math. 6 (1980) 121–125] Siemaszko generalizes for branched continued fractions the formula that expresses the difference of two successive convergents of an ordinary continued fraction. However, the generalization is not yet fit to write the branched continued fraction as an Euler-Minding series for the following reason. Indeed a convergent of the branched continued fraction can be written as a partial sum of a series but different convergents are different partial sums of different series. The next convergent cannot be obtained from the previous one by adding some terms. We shall develop here another formula that overcomes this problem.     

A Gauss-Kusmin theorem for optimal continued fractions

A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz's Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal -expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal -expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal -expansion.