A Nonvanishing Lemma for Certain Padé Approximations of the Second Kind

Abstract

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.

Blossoming and Hermite-Padé approximation for hypergeometric series

“…Quite a number of articles have addressed the question of type II Hermite-Padé approximation for various classes of hypergeometric and qhypergeometric functions, e.g., [10,30,8,9,15]. A common strategy to obtain type II Hermite-Padé approximants for either hypergeometric or q-hypergeometric series defined in a more general way than usual, was presented in [19], see also [23]. It applies in particular to (1.6) and (1.8), with however a slight limitation on the integers n 0 , n.…”

“…Though not developed here in its full generality, our blossoming approach gives further insight in the subject. Its forcefulness lies in the fact that it naturally enables us to achieve not only new explicit interpretations of existing Hermite-Padé approximants, new approximants (situations (1.7) and (1.9) depicted above), but also, simultaneously, explicit integral representations of the corresponding remainders, which is especially crucial for arithmetic applications [4,11,23]. Moreover, the fact that all the approximants are obtained as specific values of some blossoms explains Seigel's result in [28] as a straightforward consequence of the definition of blossoms.…”