Generalized Logistic Maps and Convergence

The paper explicitly treats the recurrence x_{k+1}=x_k(1-x_k)^2, states the leading asymptotics x_k ~ 1/(2k) - (3/8) ln k/k^2 - C/k^2 with C=C(x_0), and then determines the next coefficients u,v,w,p,q,r,s by inserting a log-polynomial ansatz and matching terms after expanding k→k+1 and x - 2x^2 + x^3; the values reported are u=9/32, v=(3/2)C - 9/32, w=2C^2 - (3/4)C + 5/32, p=-27/128, q=-(27/16 C - 135/256), r=-(9/2 C^2 - 45/16 C + 9/16), s=-(4 C^3 - 15/4 C^2 + 3/2 C - 51/256) (derived in the paper’s Section 1 via explicit coefficient matching) . The candidate solution reproduces the same pipeline and coefficients, with an extra rigorous step establishing the existence of C by a telescoping identity and convergence of Y_k := 1/x_k - 2k - (3/2)ln k. Both methods rely on the same asymptotic ansatz and matching; the paper defers the leading-term justification to prior work [2,3] while the model supplies it directly. Hence both are correct and substantially the same proof style, with the model adding a standard justification for the constant C while the paper cites it and then carries out the coefficient-matching calculation .