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A special class of polynomials orthogonal on the unit circle including the associated polynomials. (English) Zbl 0854.42021
Summary: Let $(P_\nu)$ be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, $(\Omega_\nu)$ the monic associated polynomials of $(P_\nu)$, and let $A$ and $B$ be self-reciprocal polynomials. We show that the sequence of polynomials $(AP_{\nu+ \lambda}+ B\Omega_{\nu+ \lambda})/ z^\lambda$, $\lambda$ suitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as the $P_\nu$’s from a certain index value onward, and determine the orthogonality measure explicitly. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order of $P_\nu$ and of the corresponding orthogonality measure and Szegö function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally, necessary and sufficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.

42C05General theory of orthogonal functions and polynomials
Full Text: DOI
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