We investigate in this paper the global behavior of the following difference equation: x n + 1 = ( P k ( x n − i 0 , x n − i 1 , … , x n − i 2 k ) + b ) / ( Q k ( x n − i 0 , x n − i 1 , … , x n − i 2 k ) + b ) , n = 0 , 1 , … , under appropriate assumptions, where b ∈ [ 0 , ∞ ) , k ≥ 1 , i 0 , i 1 , … , i 2 k ∈ { 0 , 1 , … } with i 0 < i 1 < … < i 2 k , the initial conditions x i − 2 k , x i − 2 k + 1 , … , x 0 ∈ ( 0 , ∞ ) . We prove that unique equilibrium x ¯ = 1 of that equation is globally asymptotically stable.