We investigate the asymptotic behavior of the recursive difference equation y n + 1 = ( α + β y n ) / ( 1 + y n − 1 ) when the parameters α < 0 and β ∈ ℝ . In particular, we establish the boundedness and the global stability of solutions for different ranges of the parameters α and β . We also give a summary of results and open questions on the more general recursive sequences y n + 1 = ( a + b y n ) / ( A + B y n − 1 ) , when the parameters a , b , A , B ∈ ℝ and a b A B ≠ 0 .