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Structure relations for orthogonal polynomials on the unit circle. (English) Zbl 1242.33011

Summary: Structure relations for orthogonal polynomials with respect to Hermitian linear functionals are studied. Firstly, we prove that semi-classical orthogonal polynomials satisfy structure relations of the following type: \[ \sum^{s_1}_{k=0}\beta_{n,k}P_{n+s_1-k}+\sum^{s_2}_{k=0}\gamma_{n,k}z^kP^*_{n-1-k}=\sum^{r_1}_{k=0}\alpha_{n,k}P^{[1]}_{n+s_1-k}+\sum^{r_2}_{k=0}\eta_{n,k}(P^*_{n+r_2-k})', \] where \(s_1,s_2,r_1,r_2\) are integers (specified in the text), \(P^*_n\) is the reversed polynomial of \(P_n,P^{[1]}_n=P_{n+1}'/(n+1)\), and \(\beta_{n,k},\gamma_{n,k},\alpha_{n,k},\eta_{n,k}\) are complex numbers. Then, we study the semi-classical character of sequences of orthogonal polynomials \(\{R_n\},\{P_n\}\), connected through a structure relation of the following type: \[ \sum^{s_1}_{k=0}\beta_{n,k}R_{n+s_1-k}+\sum^{s_2}_{k=0}\gamma_{n,k}R^*_{n +s_2-k}=\sum^{r_1}_{k=0}\alpha_{n,k}P^{[1]}_{n+r_1-k}+\sum^{r_2}_{k=0}\eta_{n,k}(P^*_{n+r_2-k})', \] where the integers \(s_1,s_2,r_1,r_2\) satisfy some natural conditions specified in the text.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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