Repeated modifications of limitk-periodic continued fractions

Abstract

Summary. The advantages of using modified approximants for continued fractions, can be enhanced by repeating the modification process. If K(a,/b,) is limit k-periodic, a natural choice for the modifying factors is a k-periodic sequence of right or wrong tails of the corresponding k-periodic continued fraction, if it exists. If the modified approximants thus obtained are ordinary approximants of a new limit k-periodic continued fraction, we repeat the process, if possible. Some examples where this process is applied to obtain a convergence acceleration are also given.

Improving a method for computing non-dominant solutions of certain second-order recurrence relations of Poincar�-type

“…In the third paragraph we give some examples where we use our method to calculate minimal solutions of linear homogeneous second-order recurrence relations (see [I, 5, 11]). In the fourth paragraph we prove that the method is equivalent to repeated applications of the Bauer-Muir transformation, and in this sense it is related to the idea developed by Jacobsen in [2]. Finally in the last paragraph we generalize these ideas to non-homogeneous recurrence relations.…”

“…A continued fraction (2) converges to w~. It is then possible to prove (Thron and Waadeland [12]) that the modified approximants S.(wl.))…”

Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions

“…We also call (4.3) a TW-transformation of (1.1) with respect to {w n } [17,19,20,27]. This transform can be used repeatedly, that is, we can apply the Bauer-Muir transformation repeatedly to the new continued fraction bi + f (i) for the (i + 1)th iteration, i = 0, 1, 2, .…”