Dynamics of the degree six Landen transformation (Q874393)

Let \(p(x)=cx^4+dx^2+e, q(x)=x^6+ax^4+bx^2+1\) and \(U_6(a,b;c,d,e)= \int_0^{\infty} p(x)/q(x) dx.\) \textit{G. Boros} and \textit{V. H. Moll} proved in [Math. Comput. 71, No. 238, 649--668 (2002; Zbl 0988.33009)] the existence of a five term recursive transformation (involving \((a,b,c,d,e)\)) which leaves \(U_6(a,b;c,d,e)\) invariant. Two of these recursive sequences involve only \(a\) and \(b\). Here the authors consider the dynamics of these two sequences and describe the region of convergence of \(U_6\) in the \((a,b)\)-plane as the basin of attraction of a fixed point.