Mixed recurrence relations and interlacing of the zeros of some \(q\)-orthogonal polynomials from different sequences
The fact that the zeros of orthogonal polynomials of degree \(n\) and \(n-1\) interlace has been studied extensively. For \(n\in\mathbb{N}\) let us denote by \(p_n\) an orthogonal polynomial of degree \(n\) on the interval \((c,d)\). The author presents two generalizations of known results. Let \(f(x)=a_1(x) p_n(x) +a_0(x)p_{n-1}(x)\) be continuous. (I) If \(a_0\) and \(a_1\) are continuous and of constant sign on \((c,d)\), then \(f\) has at least one zero between any two consecutive zeros of \(p_{n-i}\). (Cf. [\textit{C. Brezinski}, \textit{K. A. Driver} and \textit{M. Redivo-Zaglia}, Appl. Numer. Math. 48, No. 2, 157--168 (2004; Zbl 1047.33002)].) (II). If \(f\) is a polynomial and both \(a_0\) and \(a_1\) are continuous and of constant sign on \((c,d)\), then all zeros of \(f\) are real and simple; the zeros of \(f\) and \(p_n\) interlace and those of \(f\) and \(p_{n-1}\) interlace. (Cf. [\textit{T. S. Chihara}, An introduction to orthogonal polynomials. Mathematics and its Applications. Vol. 13. New York: Gordon and Breach, Science Publishers (1978; Zbl 0389.33008), Exercise 5.4].) With the aid of these tools, the author gives interlacing properties in the following cases: 1. Al-Salam-Chihara polynomials with different parameters; 2. \(q\)-Meixner-Pollaczek polynomials with different parameters; 3. continuous \(q\)-ultraspherical polynomials with different parameters; 4. \(q\)-Laguerre polynomials with different parameters.
scientific article; zbMATH DE number 5881201
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