On k-Fibonacci Numbers with Applications to Continued Fractions

Let ((omega) over bar (n))(n is an element of N) be the sequence of k-Fibonacci numbers recursively defined by (omega) over bar (1) = 1; (omega) over bar (2) = 1; (omega) over bar (n+2) = k (omega) over bar (n+1) + (omega) over bar (n); for all n is an element of N; and m be a fixed positive integer. In this work we prove that, for almost every x is an element of (0, 1), the pattern k, k, ..., k (comprising of m-digits) appears in the continued fraction expansion x = [0, a(1), a(2), ...] with frequency (f) over cap (k,m) := (-1)(m) k/log 2 log {phi(-1)(m+1) + 1/phi(1)(m) + 1}, where phi(m) = (omega) over bar (m+1)/(omega) over bar (m), i.e., lim(n ->infinity) 1/n #{j is an element of Omega(n) : a(j+i) = k for all i is an element of Omega(m-1) boolean OR {0}} = (f) over cap (k, m), where Omega(n) := {1, 2, ..., n}.