On the Recursive Sequence x n = 1 + ∑ i = 1 k α i x n − p i / ∑ j = 1 m β j x n − q j

We give a complete picture regarding the behavior of positive solutions of the following important difference equation: x n = 1 + ∑ i = 1 k α i x n − p i / ∑ j = 1 m β j x n − q j , n ∈ ℕ 0 , where α i ,  i ∈ { 1 , … , k } , and β j ,  j ∈ { 1 , … , m } , are positive numbers such that ∑ i = 1 k α i = ∑ j = 1 m β j = 1 , and p i ,  i ∈ { 1 , … , k } , and q j ,  j ∈ { 1 , … , m } , are natural numbers such that p 1 < p 2 < ⋯ < p k and q 1 < q 2 < ⋯ < q m . The case when gcd ( p 1 , … , p k , q 1 , … , q m ) = 1 is the most important. For the case we prove that if all p i ,  i ∈ { 1 , … , k } , are even and all q j ,  j ∈ { 1 , … , m } , are odd, then every positive solution of this equation converges to a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.