On the transcendence and algebraic independence of certain continued fractions

The transcendence of continued fractions =[a ₀;a ₁,a ₂,...] is proved under growth conditions involving the denominatorsq _n of the convergents and shifted partial quotientsa _n+k . Extending this idea, conditions for the algebraic independence of several continued fractions are given. The proofs use the approximation properties of continued fractions in combination with the Thue-Siegel-Roth Theorem or a criterion for algebraic independence of Bundschuh.     

Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences

The main theorem of this paper, proved using Mahler's method, gives a necessary and sufficient condition for the values Θ(x,a,q) at any distinct algebraic points to be algebraically independent, where Θ(x,a,q) is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives Θ(x,a,q) taking algebraically independent values for any distinct triplets (x,a,q) of nonzero algebraic numbers. Moreover, Θ(a,a,q) is expressed as an irregular continued fraction and Θ(x,1,q) is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively.     

Concerning algebraic independence of some transcendental numbers

Given the three numbers of,a ^{1 2} , and Ina ₂/Ina ₁, wherea ₁ anda ₂ are algebraic numbers whose logarithms are linearly independent in a rational field and is a quadratic irrationality, it is shown that they are not all expressible algebraically in terms of one of them.Translated from Matematicheskie Zametki, Vol.3, No. 1, pp. 51–58, January, 1968.     

The degree of approximation of certain analytic continued fractions

Summary Continued fraction expansions of classes of holomorphic functions associate with each function its sequence of coefficients. If there is a probability measure on the sequence space one can investigate the almost everywhere convergence properties of the continued fraction. In this paper we compute the degree of approximation for the expansions of functions holomorphic and bounded in the disk and for the expansion of the PR functions used in network synthesis.This research was partially supported by National Science Foundation Grants GP-6852 and GP-6216.     

Representation of algebraic numbers by periodic branching continued fractions

The notion of a periodic branching continued fraction is a natural generalization of the notion of a periodic continued fraction. It is shown that for an arbitrary positive algebraic number one can construct a periodic branching continued fraction with natural elements convergent to this number.     

Continued fractions for certain algebraic power series

Let F be an arbitrary field and let K = F((x⁻¹)) be the field of formal Laurent series in x⁻¹ over F. The usual theory of continued fractions carries over to K, with the polynomials in x playing the role of the integers. We study the continued fraction expansions of elements of K which are algebraic over F(x), the field of rational functions of x.We give the first explicit expansions of algebraic elements of degree greater than 2 for which the degrees of the partial quotients are bounded. In particular we give explicitly the continued fraction expansion for the solution f in K of the cubic equation xf³ + f + x = 0 when F = GF(2). This cubic was studied by Baum and Sweet. We give examples, for every field F of characteristic greater than 2, of algebraic elements of degree greater than 2 whose partial quotients are all linear, and we give these expansions explicitly. These are the first known examples with partial quotients of bounded degree when F has characteristic greater than 2.     

The algebraic independence of certain numbers related by the exponential function

We give a variation of a theorem of Gelfond. One of the corollaries is Schneider's conjecture that either exp(e) or exp(e²) is transcendental. It also follows that at least one of the numbers exp(α), exp(α²), exp(α³) is transcendental for any nonzero complex number α.     

Extremal values of continuants and transcendence of certain continued fractions

We prove a criterion for the transcendence of continued fractions whose partial quotients are contained in a finite set {b₁,…,b_r} of positive integers such that the density of occurrences of b_i in the sequence of partial quotients exists for 1ir. As an application we study continued fractions [0,a₁,a₂,a₃,…] with a_n=1+([nθ]modd) where θ is irrational and d2 is a positive integer.     

Determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators

An explicit construction of a reduced hyperbolic integer operator from the group SL(2, ?) such that one of the periods of the corresponding geometric continued fraction in the sense of Klein coincides with a given sequence of positive integers is presented. An algorithm determining periods for any operator in SL(2, ?) (which is based on Gauss’ reduction theory) is experimentally studied.     

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.     

Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

The Ramanujan Journal - Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p-adic numbers $$mathbb Q_p$$ . Here, we...