Algebraic independence of the values of certain series by Mahler's method

Suppose thatf ₁(z), ...f _m(z) are algebraically independent functions of a complex variable satisfying $$f_i (z) = a_i (z)f_i (Tz) + b_i (z),$$ wherea _i (z),b _i (z) are rational functions andTz=p(z ^?1)^?1 for a polynomialp(z) of degree larger than 1. We show thatf ₁(a), ...,f _m (a) are algebraically independent under suitable conditions onf anda. As an application of our main result, we deduce three corollaries, which generalize earlier work by Davison and Shallit and by Tamura.     

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 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     

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 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 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 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 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 some values of the exponential function

We prove general results concerning the algebraic independence of three values of the exponential function. Forβ algebraic and of degree 7 andα algebraic and ≠ 0, 1 there exist among the numbers α_β,..., \(\alpha ^{\beta ^6 } \) three which are algebraically independent. The proof employs a method due to A. O. Gel'fond and N. I. Fel'dman.     

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.     

Fourier series on certain solenoids

Mathematische Annalen -     

Algebraic independence of values of a subclass of entire functions

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