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.

     

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.     

Algebraic independence of the values of Mahler functions associated with a certain continued fraction expansion

It is proved that the function , which can be expressed as a certain continued fraction, takes algebraically independent values at any distinct nonzero algebraic numbers inside the unit circle if the sequence {R_k}_k?0 is the generalized Fibonacci numbers.     

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 theory of difference equations and Mahler functions

Algebraic independence of certain Mahler functions constructed from Rudin–Schapiro sequences and Baum–Sweet sequences is proved, using difference Riccati equations and the notion of difference field extension of valuation ring type.     

Algebraic independence of values of exponential and elliptic functions

With the aid of Gel'fond's method [1, 2], which makes it possible to show that there exist at least two algebraically independent quantities among several values of the exponential function, and by using certain additional considerations, the author obtains a result concerning the algebraic independence of the values of the exponential and the elliptic functions.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 195–202, August, 1976.     

Algebraic independence of logarithmic singularities of some complex functions

We show that algebraic independence of some complex functions of one variable over regular functions implies their algebraic independence over a larger ring, containing complex powers of regular functions. Based on this we obtain a generalization of a special case of the theorem of Kaczorowski and Perelli on functional independence of logarithms of functions in the Selberg class. As an application we state a new result on oscillations of arithmetical functions.     

Algebraic independence of values of a subclass of entire functions

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

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.     

Algebraic independence of exponents

Three theorems are obtained for the algebraic independence of some numbers related to exponential functions. Theorems 1 and 3 are extensions of the well-known Gelfond results.     

Algebraic Independence of Certain Generalized Mahler Type Numbers

In this paper the generalized Mahler type number Mh（g;A,T） is defined, and in the case of multiplicatively dependent parameters gi, hi（1 ≤ i ≤ s） the algebraic independence of the numbers Mhi （gi; A, T）（1 ≤ i ≤ s） is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.     

Algebraic independence of certain numbers

In this note the algebraic independence of values of general Mahler series with different parameters at algebraic points is established. Supported by the Fok Ying Tung Education Foundation and the National Natural Science Foundation of China     

Mahler型函数值的代数无关度量

In this paper,we give the algebraic independence measures for the values of Mahler type functions in complex number field and p-adic number field,respectively.     

Algebraic independence by approximation method

By the use of approximation method a general criterion of algebraic independence of complex numbers is established. As its application the algebraic independence of values of certain gap series at algebraic and transcendental points is given. Subject supported by the National Natural Science Foundation of China     

Mahler型函数值的代数无关度量

利用初等的结式方法研究满足多项式形式的函数方程组的Mahler型函数的零点估计,给出了满足非线性函数方程组的Mahler型函数在代数点值的代数无关度量.     

Mahler型函数值的代数无关度量

利用初等的结式方法研究满足多项式形式的函数方程组的Mahler型函数的零点估计,给出了满足非线性函数方程组的Mahler型函数在代数点值的代效无关度量．