Arithmetical properties of the solutions of certain functional equations
by
Peter Bundschuh
Universität Köln

In our talk, we shall mainly consider entire transcendental functions f satisfying linear q-difference equations, i.e. equations of Poincaré type f(q^mz) = R₀(z)f(q^m-1z)+ ... +R_m-1(z)f(z)+R_m(z), where m is a positive integer, q a fixed complex number with |q| > 1, and R₀, ... , R_m are polynomials. We shall discuss several analytic methods to prove irrationality of f(\alpha) or linear independence of f(\alpha₀), ... , f(\alpha_n) at appropriate points \alpha or \alpha₀, ... , \alpha_n, or, more generally, to estimate from below the dimension of the vector space Qf(\alpha₀)+ ... +Qf(\alpha_n) over Q. Apart from these purely qualitative questions, we shall be interested in their quantitative refinements as well.

Of course we will give applications of the results to certain, now ``classical'' special functions f of the above type. Finally, we shall point out some, at the moment quite singular, transcendence results in this field, which became available after Nesterenko's famous work from 1996 on modular functions.

Date received: April 19, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-39.