Keijo Väänänen

An application of Jacobi type polynomials to irrationality measures

The paper provides irrationality measures for certain values of binomial functions and definite integrals of some rational functions. The results are obtained using Jacobi type polynomials and divisibility considerations of their coefficients.

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

New irrationality measures for q-logarithms

The three main methods used in diophantine analysis of q-series are combined to obtain new upper bounds for irrationality measures of the values of the q-logarithm function when p = 1/q ∈ Z \ {0, ±1} and z ∈ Q.

ON IRRATIONALITY MEASURES OF THE VALUES OF GAUSS HYPERGEOMETRIC FUNCTION

The paper gives irrationality measures for the values of some Gauss hypergeometric functions both in the archimedean andp-adic case. Further, an improvement of general results is obtained in the case of logarithmic function.

Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes

Two recursive algorithms for computing the weight distributions of certain binary irreducible cyclic codes of length n in the so-called index 2 case are presented. The running times of these algorithms are smaller than O(log/sup 2/r) where r=2/sup m/ and n is a factor of r-1.

Baker-type estimates for linear forms in the values of q-series

We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field II, in particular of the values of q-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegels method applied to a system of functional Poincare-type equations and the connection between the solutions of these functional equations and the generalized Heine series.

Zu einem Satz von Skolem über lineare Unabhängigkeit von Werten gewisser Thetareihen

We consider arithmetic properties of the solution of a functional equation f(qx)=xf(x)+1, where q≠0 is a rational number satisfying certain conditions. A linear independence measure is obtained for the values of f and its derivatives at rational points, both in the archimedian and p-adic case. The proof uses a refinement of the method followed by skolem in the corresponding qualitative considerations.

On rational approximations of certain Mahler functions with a connection to the Thue–Morse sequence

We shall obtain the irrationality exponent 2 for some values of two special Mahler functions. This gives a new proof for the recent result of Bugeaud on Thue–Morse–Mahler numbers.

On the non-quadraticity of values of the

We investigate arithmetic properties of values of the entire function