Algebraic independence of certain Mahler functions

A monotonicity-type result for functions \(f\ : \ \mathbb {N}_a\rightarrow \mathbb {R}\) satisfying the sequential fractional difference inequality

$$\begin{aligned} \Delta _{1+a-\mu }^{\nu }\Delta _{a}^{\mu }f(t)\ge 0, \end{aligned}$$

for \(t\in \mathbb {N}_{2+a-\mu -\nu }\), where \(0<\mu <1\), \(0<\nu <1\), and \(1<\mu +\nu <2\), is proved, subject to the restriction that

$$\begin{aligned} \mu <2(1-\nu ). \end{aligned}$$

We demonstrate that this result is sharp in the sense that the restriction \(\mu <2(1-\nu )\) cannot be improved.

     

Algebraic independence of reciprocal sums of binary recurrences

Algebraic independence of the numbers , where{R _n } _n 0 is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler's method.     

Mahler函数值的代数无关性

给出M ah ler型函数值超越次数的下界.     

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.     

Algebraic independence of Mahler functions

Suppose that f₁, ?, f_mf_1, \ldots , f_m satisfy functional equations of type¶¶ f_i(z^d) = P_i(z, f_i(z))     or     f_i(z) = P_i(z, f_i(z^d))f_i({z^d}) = P_i(z, f_i(z)) \quad {or} \quad f_i(z) = P_i(z, f_i({z^d})) ¶for i = 1, ?, mi = 1, \ldots , m, an integer d > 1 and polynomials P_i ? \Bbb C (z)[ y]P_i \in \Bbb C (z)[ {y}] with pairwise distinct partial degrees deg_y( P₁), ?, deg_y( P_m)\deg _y( {P_1}), \ldots , \deg _y( {P_m}). Generalizing a result of Keiji Nishioka and using an idea of Kumiko Nishioka we show, that f₁, ?, f_mf_1, \ldots , f_m can only be algebraically dependent over \Bbb C (z)\Bbb C (z), if there is an index k ? { 1, ?, m}\kappa \in \{ {1, \ldots , m}\} such that f_kf_{\kappa } is rational.     

Algebraic independence results for the values of certain Mahler functions and their application to infinite products

In this paper we establish algebraic independence criteria for the values at an algebraic point of Mahler functions each of which satisfies either a multiplicative type of functional equation or an additive one. As application we construct, using a linear recurrence sequence, an entire function defined by an infinite product such that its values as well as its all successive derivatives at algebraic points other than its zeroes are algebraically independent. Zeroes of such an entire function form a subsequence of the linear recurrence sequence. We prove the algebraic independency by reducing those values at algebraic points to those of Mahler functions.     

Algebraic independence of arithmetic gamma values and Carlitz zeta values

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq[θ] and provide complete algebraic independence results for them.     

Algebraic independence of values of a certain class of hypergeometric functions

We prove a general theorem on the algebraic independence of values of hypergeometric E-functions and their successive derivatives at algebraic points for the degenerate case in which substantial cancellations occur in numerators and denominators of coefficients of the series in powers ofz of the functions considered.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 402–414, March, 1996.     

Linear independence of digamma function and a variant of a conjecture of Rohrlich

Let ψ(x) denote the digamma function. We study the linear independence of ψ(x) at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series.     

Rational functions over finite fields having continued fraction expansions with linear partial quotients

Let F be a finite field with q elements and let g be a polynomial in F[X] with positive degree less than or equal to q/2. We prove that there exists a polynomial f∈F[X], coprime to g and of degree less than g, such that all of the partial quotients in the continued fraction of g/f have degree 1. This result, bounding the size of the partial quotients, is related to a function field equivalent of Zaremba's conjecture and improves on a result of Blackburn [S.R. Blackburn, Orthogonal sequences of polynomials over arbitrary fields, J. Number Theory 6 (1998) 99-111]. If we further require g to be irreducible then we can loosen the degree restriction on g to deg(g)?q.     

Transcendence of certain infinite products

We prove the transcendence results for the infinite product , where Ek(x), Fk(x) are polynomials, α is an algebraic number, and r?2 is an integer. As applications, we give necessary and sufficient conditions for transcendence of and , where Fn and Ln are Fibonacci numbers and Lucas numbers respectively, and {ak}_k?0 is a sequence of algebraic numbers with log‖ak‖=o(rk).     

Linear independence of automatic formal power series

We extend a result of J.-P. Allouche and O. Salon on linear independence of formal power series associated to polynomial extractions of quasistrongly p-additive sequences. The original result was on the F_p-linear independence and we extend it to the F_p[X]-linear independence.     

Transcendence of certain series involving binary linear recurrences

Duverney and Nishioka [D. Duverney, Ku. Nishioka, An inductive method for proving the transcendence of certain series, Acta Arith. 110 (4) (2003) 305-330] studied the transcendence of , where Ek(z), Fk(z) are polynomials, α is an algebraic number, and r is an integer greater than 1, using an inductive method. We extend their inductive method to the case of several variables. This enables us to prove the transcendence of , where Rn is a binary linear recurrence and {ak} is a sequence of algebraic numbers.     

Transcendental values of the digamma function

Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler's Γ-function. Let q be a positive integer greater than 1 and γ denote Euler's constant. We show that all the numbers

     

Algebraic independence of values of certain Fourier series

The algebraic independence of values of certain Fourier series with algebraic coefficients and their derivatives at algebraic points is proved.     

Algebraic independence of values of a subclass of entire functions

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 46–51, June, 1992.     

The theta group and the continued fraction expansion with even partial quotients

F. Schweiger introduced the continued fraction with even partial quotients. We will show a relation between closed geodesics for the theta group (the subgroup of the modular group generated by z+2 and -1 / z) and the continued fraction with even partial quotients. Using thermodynamic formalism, Tauberian results and the above-mentioned relation, we obtain the asymptotic growth number of closed trajectories for the theta group. Several results for the continued fraction expansion with even partial quotients are obtained; some of these are analogous to those already known for the usual continued fraction expansion related to the modular group, but our proofs are by necessity in general technically more difficult.Supported by The Netherlands Organization for Scientific Research (NWO).     

Algebraic Independence of Reciprocal Sums of Binary Recurrences II

 Algebraic independence of the numbers for various d and l, where is a periodic sequence of algebraic numbers and is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler’s method. Received 25 August 2000; in revised form 8 January 2002     

Convergence of continued fraction type algorithms and generators

The concept of convergence of continued fraction type algorithms has been defined a number of times in the literature. We investigate the relation between these definitions, and show that they do not always coincide. We relate the definitions to the question whether or not the natural partition of the underlying dynamical system is a generator. It turns out that the right definition of convergence is equivalent to this partition being a generator. The second definition of convergence is shown to be equivalent only under extra conditions on the transformation. These extra conditions are typically found to be satisfied when the second definition is used in the literature.