A canonical family of multiple orthogonal polynomials for Nikishin systems
Let \((\mu_1,\mu_2)=N(\sigma)\) be a Nikishin system of measures, generated by a pair of measures \(\sigma=(\sigma_1,\sigma_2)\), supported on compact disjoint intervals \(\Delta_1\), \(\Delta_2\). It is well known that if \(\sigma_k'>0\) a.e. on \(\Delta_k\), \(k=1,2\), then the sequence \(\{Q_n\}_{n\geq0}\) of monic multiple orthogonal polynomials for \(N(\sigma)\) satisfies the four-term recurrence relation with asymptotically \(2\)-periodic coefficients. The authors prove that any sequence of monic polynomials \(\{P_n\}_{n\geq0}\) defined by recurrences with \(2\)-periodic constant coefficients \[ \begin{aligned} zP_{2n}(z) &=P_{2n+1}(z)+\alpha_0 P_{2n}(z)+\beta P_{2n-1}(z)+\gamma_0 P_{2n-2}(z),\\ zP_{2n+1}(z) &=P_{2n+2}(z)+\alpha_1 P_{2n+1}(z)+\beta P_{2n}(z)+\gamma_1 P_{2n-1}(z), \end{aligned} \] \(n=0,1,\dots\), with initial conditions \(P_0=1\), \(P_{-1}=P_{-2}=0\) is the sequence of monic multiple orthogonal polynomials with respect to a certain Nikishin system \(N(\sigma)\), and study the properties of this system.
scientific article; zbMATH DE number 5787096
scientific article