Generalized Brouncker's continued fractions and their logarithmic derivatives

TL;DR

This paper generalizes Brouncker's continued fraction to a new family y(s,r), extends derivative formulas, studies asymptotics, and presents Ramanujan formula generalizations.

Contribution

It introduces a new continued fraction y(s,r), extends derivative formulas from Brouncker's fraction, and explores asymptotic behavior and Ramanujan formula generalizations.

Findings

Derived formulas for first and second logarithmic derivatives of y(s,r)

Established asymptotic series for y(s,r) at infinity

Presented generalizations of Ramanujan's formulas

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.