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Some results on biorthogonal polynomials. (English) Zbl 0815.42012

Starting from the Christoffel determinant formula that gives an expression for the orthogonal polynomials that arise from polynomial modification of a weight function, the author gives a biorthogonal pair \(\{\psi_ n(z)\}\), \(\{\phi_ n(z)\}\) on the unit circle with respect to the measure \[ d\nu(\theta)= w(z) d\theta= z^{-m} (z- \alpha_ 1)(z- \alpha_ 2)\cdots (z-\alpha_ h) d\theta,\;z= e^{i\theta},\;\alpha_ j\neq 0. \] Specifying \[ w(z)= {(qz; q^ 2)_{\infty} (qz^{-1}; q)_{\infty}\over {(aqz; q^ 2)_{\infty} (bqz^{-1}; q^ 2)_{\infty}}}, \]
the author recovers a result by P. I. Pastro [J. Math. Anal. Appl. 112, 517-540 (1982; Zbl 0582.33010)].
Finally, the author turns to a measure which is necessarily positive on the unit circle, but for which there exists nevertheless a unique pair of biorthogonal sets of polynomials on the unit circle (in order to achieve this certain Toeplitz determinants have to be non-zero). Now the modification uses so-called Laurent polynomials of the special form \(z^{-m} G_{2m}(z)\) and \(z^{-(m+1)} G_{2m+ 1}(z)\) with \(G_{2m}\), \(G_{2m+ 1}\) polynomials of exact degree \(2m, 2m+1\), respectively, non- vanishing for \(z= 0\). In order to have the right degree pattern for the biorthogonal polynomials certain minors of determinants in the paper have to be non-zero.
None of the determinants governing existence and uniqueness is given explicitly in the paper.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 0582.33010
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