Two Families of Cremona Maps and orthogonal Krall-Jacobi Polynomials

Two infinite families of Cremona maps depending on one real parameter are given. For all integers $n\ge 1$ the first family of Cremona maps consists of group elements in $Bir\left({\mathbb{P}}^n\right)$ with bidegree $(n,n)$, the second family of Cremona maps consists of group elements in $Bir\left({\mathbb{P}}^{2n}\right)$ with bidegree $(2n,2n)$. For the first family and $n\ge 5$, for the second family and $n\ge 3$ the existence of this group elements and the properties depend on a conjecture. But computational results suggest that the conjecture is true for all $n$.