Metrical Property For Gcf(Is An Element Of) Expansion With The Parameter Function Is An Element Of(Kappa) = C(Kappa+1)

This paper is concerned with the metric properties of the generalized continued fractions (GCF(is an element of)) with the parameter function epsilon(k(n)), where k(n) is the nth partial quotient of the GCF(epsilon) expansion. When - 1 < epsilon(k(n)) <= 1, Zhong [ Metrical properties for a class of continued fractions with increasing digits, J. Number Theory 128 ( 2008) 1506-1515] obtained the following metrical properties:lim(n ->infinity) k(n)(1/n) (x) = e; lim(n ->infinity) (k(1)k(2) ... k(n))(1/n2) = root e lambda a.e.which are entirely unrelated to the choice of epsilon(k(n)) is an element of(-1, 1]. Here we deal with the case of epsilon(k) = c(k + 1) with constant c is an element of (0, infinity). It is proved that:lim(n ->infinity) k(n)(1/n) (x) = (1+c)(1+c/c); lim(n ->infinity) (k(1)k(2) ... k(n))(1/n2) = (1+c)(1+c/2c) lambda a.e.which change with the real c is an element of(0, infinity). Note that (1+c)(1+c/c) -> e as c -> 0. it indicates that when c -> 0, the GCF(is an element of) has the same metrical property as the case of -1 < epsilon(k(n)) <= 1