On a class of biorthogonal polynomials on the unit circle (Q5962958)

The authors study a biorthogonal system \(\{P_n,Q_n\}\), introduced by \textit{R. Askey} [``Discussion of Szegő's paper ``Beiträge zur Theorie der Toeplitzschen Formen'''', in: R. Askey (ed.), G. Szegő, Collected works. Vol. I, Boston, MA: Birkhäuser. 303--305 (1982)].NEWLINENEWLINEThe system is given by NEWLINE\[NEWLINEP_n(z;\alpha,\beta)={}_2F_1(-n,\alpha+\beta+1; 2\alpha+1; 1-z),\;Q_n(z;\alpha,\beta)=P_n(z;\alpha,-\beta),NEWLINE\]NEWLINE which is biorthogonal with respect to a complex weight \(\omega(\theta)=(1-e^{i\theta})^{\alpha+\beta}(1-e^{-i\theta})^{\alpha-\beta}\), as given below NEWLINENEWLINE\[NEWLINE{1\over 2\pi}\,\int_{-\pi}^{\pi}\,P_n(e^{i\theta};\alpha,\beta)Q_m(e^{-i\theta};\alpha,\beta)\omega(\theta)d\theta = {\Gamma(2\alpha+1)\over \Gamma(\alpha+\beta+1)\Gamma(\alpha-\beta+1)}\,{n!\over (2\alpha +1)_n}\,\delta_{n,m}.NEWLINE\]NEWLINE Several proofs of this formula were given in the literature and the authors refer the reader for historical considerations to [\textit{N. M. Temme}, Constr. Approx. 2, 369--376 (1986; Zbl 0601.41031)]. In that paper a `first' asymptotic formula was given.NEWLINENEWLINEThe main results of the paper under review are:NEWLINENEWLINETheorem 1. An explicit asymptotic formula for \(P_n(e^{i\theta};\alpha,\beta)\) where \(\operatorname{Re}(\alpha+\beta)>-1\), \(\operatorname{Re}(\alpha-\beta)\geq 0\) (\(\alpha,\beta\) complex numbers).NEWLINENEWLINETheorem 2. An electrostatic model for the zeros of the para-orthogonal polynomials NEWLINE\[NEWLINEB_n\left( z;{(\alpha-\beta)_{n+1}\over (\alpha+\beta)_{n+1}}\right)={(2\alpha)_n\over (\alpha+\beta)_n}\,{}_2F_1(-n,\alpha+\beta; 2\alpha;1-z),\;\alpha\not= 0,NEWLINE\]NEWLINE where \(\alpha\in\mathbb{R}\), \(\alpha>-1/2\) and \(i\beta\in\mathbb{R}\). These zeros solve the minimum problem for NEWLINE\[NEWLINEE(\theta_1,\ldots,\theta_n)=\sum_{k\not= j}\,\ln{1\over |e^{i\theta_k}-e^{i\theta_j}|}+p\sum_{j=1}^n\,\ln{1\over |1-e^{i\theta_j}|}+q\sum_{j=1}^n\theta_j,\;\theta_j\in (0,2\pi),NEWLINE\]NEWLINE where \(p\not= 0,q\) are two real numbers and \(n\geq 2\) unit masses are at the variable points \(\{e^{i\theta_1}, \ldots, e^{i\theta_n\}}\) and one fixed mass point \(p\) at \(z=+1\).