Convergence of Continued Fractions
by
Lisa Lorentzen
Norwegian University of Science and Technology

In this survey talk we shall mainly consider the following questions.

1.

Let K(a_n(z)/b_n(z)) be a continued fraction expansion of a function f(z). How can we determine whether K(a_n(z)/b_n(z)) converges to f(z)?

2.

If {f_k(z)} satisfies the recurrence

fk+1(z)=bk(z)fk(z)+ak(z)fk-1(z)   for   k=1, 2, 3, ...,

we have

fk+1(z) fk(z) =bk(z)+ ak(z) fk(z)/fk-1(z)

and thus

- f1(z) f0(z) = a1(z) b1(z)- f2(z) f1(z) = a1(z) b1(z)+ a2(z) b2(z)- f3(z) f2(z)    etc.

How can we determine whether K(a_n(z)/b_n(z)) is an expansion of -f₁(z)/f₀(z)?

3.

The approximants of for instance a C-fraction K(a_nz^a_n/1) are rational functions. How can one avoid that the zeros of the denominators ruin the convergence of K(a_nz^a_n/1)?

4.

Continued fraction expansions can be applied to extend the region of convergence in the sense that the continued fraction may converge in a larger domain than the corresponding power series. But when do we get this benefit?

Date received: March 21, 2000

Copyright © 2000 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 # caew-46.