On Arithmetical Properties of Certain q-Series

Abstract

By the use of second type Padè approximations combined with a qiteration process, linear independence results are obtained for functions fs(z) and f s (z) defined by the seriesat points z = ±q j , 1 ≤ j ≤ s. The function values in turn correspond to qanalogues of some well known constants such as π, log 2 and ζ(2), the Riemann zeta function at point 2.

On the linear independence of the values of Gauss hypergeometric function

“…a certain determinant formed by the approximation polynomials does not vanish identically. This is the most important contribution of this work since the method of proof is new, and it applies also to the case of Heine series considered in [22]. The essential ingredient of the Padé construction, the application of the binomial (or q-binomial) theorem, is also the foundation of the determinant consideration.…”

“…, D, in the spirit of Stihl [31], an approach originating from Maier [19] and thereafter generalized by Chudnovsky [7]. See also [22], where linear independence is considered for values of functions, and their derivatives, deriving from the Heine series, a q-analogue of the Gauss series.…”