In this note we investigate arithmetic properties of values of the Tschakaloff function Tq(z) = P∞ ν=0 q −ν(ν+1)/2zν , |q| > 1. One of the open problems is proving linear independence of the values of Tq(z) with different q. The only result obtained in this direction in [1] is very restrictive. We refer an interested reader to the survey [2] for an account of known linear and algebraic independ...

An expression in terms of (cyclotomic) harmonic sums can be simplified by the quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sums are algebraically independent within the quasi-shuffle algebra. In this article we show that the basis sums can be represented within a tower of difference ring extensions where the constants remain unchanged. This property enables...

In this paper, the algebraic independence of values of the functionGd(z) := ∑ h≥0 z dh/(1− zdh ), d>1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly interesting case of multiplicatively dependent points is considered here, and similar results are obtained for more ...

Uncertain field is virtually an extension of uncertain process with the index space changing from a totally ordered set into a partially ordered one such as time-space or a surface. For describing the uncertain field, this paper introduces several concepts of uncertainty distribution and inverse uncertainty distribution. In addition, a sufficient and necessary condition is proved for the uncert...

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable. Such domains turn out to have the ...

In C-tests the hypothesis of items local independence is violated, which doesn't permit to consider them as real tests. It is suggested to determine the distances between separate C-test items (blanks) and to combine items into clusters. Weights, inversely proportional to the number of items in corresponding clusters, are assigned to items. As a result, the C-test structure becomes similar to t...