  We investigate determinantal point processes on $[0,+\infty)$ of the form
\begin{equation*}\label{probability distribution} \frac{1}{Z_n}\prod_{1\leq
i<j\leq n}(\lambda_j-\lambda_i)\prod_{1\leq i<j\leq
n}(\lambda_j^\theta-\lambda_i^\theta) \prod_{j=1}^n
w(\lambda_j)d\lambda_j,\qquad \theta\geq 1. \end{equation*} We prove that the
biorthogonal polynomials associated to such models satisfy a recurrence
relation and a Christoffel-Darboux formula if $\theta\in\mathbb Q$, and that
they can be characterized in terms of $1\times 2$ vector-valued Riemann-Hilbert
problems which exhibit some non-standard properties. In addition, we obtain
expressions for the equilibrium measure associated to our model if
$w(\lambda)=e^{-nV(\lambda)}$ in the one-cut case with and without hard edge.
