Truncation Error Bounds For Limit K-Periodic Continued Fractions

Abstract

Many special functions of mathematical physics have representations in the form of limit k-periodic continued fractions f= K (1/b _n), where lim_{n→ ∞} b _n = b or lim _{n→ ∞} b _kn +i = β _i where k is a positive integer and 0 ≤ i ≤ k — 1. For computation of such functions by continued fractions, one needs sharp bounds for the truncation error $\left| {f - {f_n}} \right|$ resulting when the value f is replaced by the nth approximate f _n. In this paper we present bounds for limit 4-periodic continued fractions K(1/b_n). Some of these bounds are shown to be “best” relative to limited given information.

Research supported in part by the National science Foundation under Grant No. DMS-9302584