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.


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.)


Zbl 0582.33010
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