On the acceleration of limit periodic continued fractions

Summary

A continued fraction (c.f.)K(a _n /1) is called limit periodic if$\mathop {\lim }\limits_{n \to \infty } a_n = a$. Fora∈ℂ anda∉(−∞,−1/4],a≠0, Thron-Waadeland (1980) examined a modification of a limit periodic c.f. for accelerating the convergence. This acceleration remains modest if thea _n converge only logarithmically. Thus it is proposed to apply an Euler summability method to the series equivalent to the c.f. Properties of the “equivalent function” are derived. These properties are used for choosing appropriate parameters for the summability method such that a considerable acceleration can be expected even if thea _n converge logarithmically.