Generalized Brouncker's continued fractions and their logarithmic derivatives

Abstract

In this paper, we study the continued fraction y(s,r) which satisfies the equation y(s,r)y(s+2r,r)=(s+1)(s+2r-1) for r > 1/2. This continued fraction is a generalization of the Brouncker's continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s,r). The asymptotic series for y(s,r) at the infinity are also studied. The generalizations of some Ramanujan's formulas are presented.