We shall consider arithmetical properties of the q-continued fractionsand some related continued fractions where v is a fixed valuation of an algebraic number field K and s, h, l ∈ N. In particular, we get sharp irrationality measures for certain Ramanujan, RamanujanSelberg, Eisenstein and Tasoev continued fractions.

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.

We prove the nonvanishing lemma for explicit second kind Padé approximations to generalized hypergeometric and q-hypergeometric functions. The proof is based on an evaluation of a generalized Vandermonde determinant. Also, some immediate applications to the Diophantine approximation is given in the form of sharp linear independence measures for hypergeometric E- and G-functions in algebraic number fields with different valuations.

Let polynomials S(t), T(t) be given, then the convergence of the q-continued fraction [Formula: see text] will be studied using the Poincaré–Perron Theorem and Frobenius series solutions of the corresponding q-difference equation S(t)H(q2t) = -T(t)H(qt) + H(t). Our applications include a generalization of a q-continued fraction identity of Ramanujan and certain q-fractions, which arise in the theory of q-orthogonal polynomials.

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