On three term recurrence relations for orthonormal polynomials on the unit circle
Let \(\{P_ n(z)\}\) be the system of orthogonal polynomials on the unit circle with respect to a given weight function \(w(z)\). Further, let \(P_ n(z)=\sum^ n_{h=0}\alpha_{h,n}z^ h\) and \(k_ n=\alpha_{n,n}\). The authors prove that if \(n\geq 2\) and \(\alpha_{0,n-1}\neq 0\), then \[ A_ nP_ n(z)=(z+B_ n)P_{n-1}(z)-A_{n-1}B_ nzP_{n-2}(z), \] where \(A_ n=k_{n-1}/k_ n\), \(B_ n=(\alpha_{0,n}/\alpha_{0,n- 1})A_ n\). Analogous recurrence relations are given for the polynomials \(P^*_ n(z)=\sum^ n_{h=0}\overline\alpha_{h,n}z^{n-h}\), \(P_{-n}(z)=\sum^ n_{h=0}\overline\alpha_{h,n}z^{-h}\) and \(P^*_{-n}(z)=\sum^ n_{h=0}\alpha_{h,n}z^{n-h}\).
polynomials orthogonal on the unit circle
scientific article; zbMATH DE number 91673