On the polynomials orthogonal on regular polygons
The author studies the so called Pastro-Al Salam-Ismail polynomials [\textit{W. A. Al-Salam} and \textit{M. E. H. Ismail}, Proc. Am. Math. Soc. 121, 553-561 (1994; Zbl 0835.33011) and J. Math. Anal. Appl. 112, No. 2, 517-540 (1985; Zbl 0582.33010)]: polynomials biorthogonal on the unit circle with continuous weight function (under mild conditions). They satisfy for \(n\in \mathbb{Z}\) \[ P_{n+1}(z)=zP_n(z)+q^{(n+1)/2}{(b;q)_{n+1}\over (aq;q)_{n+1}}Q_n(1/z), \] \[ Q_{n+1}(z)=zQ_n(z)+q^{(n+1)/2}{(a;q)_{n+1}\over (bq;q)_{n+1}}P_n(1/z), \] with \(P_0(z)=Q_0(z)=1\). Here \(a,b\) are arbitrary complex numbers, \(q\) is real with \(|q|<1\) and the \(q\)-shifted factorial is defined as usual: \((a;q)_n=(1-a)(1-aq)\cdots (1-aq^{n-1})\). Explicit expressions for \(P_n(z;a,b)\) and \(Q_n(z;a,b)=P_n(z;b,a)\) are known. The author now takes \(q\) to be a primitive \(N\)th root of unity \(q=\exp{2\pi i/N}\) and restricts himself to \(b=aq^j\) \((j=0,1,\ldots ,N-1)\) and \(a^N,b^N,(ab)^N\not=1\). As now \(P_N(z)=z^N-(1-b^N)/(1-a^N)\), the zeros of \(P_n,Q_n\) coincide with the roots of unity and the author succeeds in calculating the weight function explicitly. Moreover, the special case \(a=b=q^{\gamma},-1/2<\gamma <1/2\) is considered, leading to a regular \(N\)-gon as the support for the weight. Finally an exceptional case \(ab=q^{-k},\;k=2,3,\ldots,N-1\) with \(b\in \mathbb{C},\;b^N\not= 1\) is studied. Now the support of the weight consists of the \(k\) vertices \(bq^{s+1/2} (s=0,1,\ldots,K-1)\) of a regular \(N\)-gon with radius \(|b|\).
scientific article; zbMATH DE number 1286384
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