Peter Bundschuh

Linear Independence of q-Analogues of Certain Classical Constants

Let K be ℚ or an imaginary quadratic number field, and q ∈ K an integer with ¦q¦ > 1. We give a quantitative version of Σn≥1 an/(qn − 1) ∉ K for non-zero periodic sequences (an) in K of period length ≤ 2. As a corollary, we get a quantitative version of the linear independence over K of 1, the q-harmonic series, and a q-analogue of log 2. A similar result on 1, the q-harmonic series, and a q-analogue of ζ(2) is also proved. Mathematics Subject Classification (2000): 11J72, 11J82

Zur Transzendenz gewisser Reihen

From Schmidts simultaneous approximation theorem we deduce transcendence results concerning series of rational numbers. The denominators of these numbers are from finitely many linear recursive sequences and have to satisfy a divisibility as well as a growth condition. (In an appendix the second author studies the connections between these two kinds of hypothesis.) For the numerators we need some growth conditions too. We study also the implications of Mahlers analytic transcendence method from 1929 to the arithmetical questions considered mainly.

Transcendental continued fractions

Abstract In two previous papers Nettler proved the transcendence of the continued fractions A := a 1 + 1 a 2 + 1 a 3 + ⋯ , B := b 1 + 1 b 2 + 1 b 3 + ⋯ as well as the transcendence of the numbers A + B, A − B, AB, A B where the as and bs 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 AB is established.

Zwei Bemerkungen über transzendente Zahlen

AbstractIn the first part theorems ofBaker are used to prove the transcendence of special values of power series whose coefficients are values of certain ordinary Dirichlet series with coefficients forming a periodic sequence of algebraic numbers. Especially the transcendence of Ψ(z)+C is shown for all rationalz which are not integers, Ψ denoting the logarithmic derivative of the gamma function andC Eulers constant. In the second part the intimate connection betweenSchanuels conjecture and the arithmetic nature of

On theorems of Gelfond and Selberg concerning integral-valued entire functions

\sum\limits_{n = 2}^\infty {\gamma (n)n^{ - s} }

Algebraic Independence of Infinite Products and Their Derivatives

,s=3,4,5... is studied where γ (n) denotes the number of distinct representation ofn in the formab with positive integersa, b. This function was recently introduced byGolomb.

Über lineare Unabhängigkeit

In this same journal, Coons published recently a paper [The transcendence of series related to Sterns diatomic sequence, Int. J. Number Theory6 (2010) 211–217] on the function theoretical transcendence of the generating function of the Stern sequence, and the transcendence over ℚ of the function values at all non-zero algebraic points of the unit disk. The main aim of our paper is to prove the algebraic independence over ℚ of the values of this function and all its derivatives at the same points. The basic analytic ingredient of the proof is the hypertranscendence of the function to be shown before. Another main result concerns the generating function of the Stern polynomials. Whereas the function theoretical transcendence of this function of two variables was already shown by Coons, we prove that, for every pair of non-zero algebraic points in the unit disk, the function value either vanishes or is transcendental.

ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

For each s ∈ N define the constant 0s with the following properties: if an entire function g(z) of type t(g) 0 there exists an entire transcendental function g(z) satisfying the display condition and t(g) < 0s + δ. The result 01 = log 2 is known due to Hardy and Polya. We provide the upper bound 0s ≤ πs/3 and improve earlier lower bounds due to Gelfond (1929) and Selberg (1941).