A class of orthogonal functions given by a three term recurrence formula (Q2796021)

Let \(\mathbb P_m\) be the linear space of real polynomials of degree at most \(m\) and let \(\Omega_m\) be the linear space of functions on \([-1,1]\) such that: \(\Omega_0=\mathbb P_0\) and \(\Omega_m\), for \(m\geq1\), is such that if \(\mathcal F\in\Omega_m\), then \(\mathcal F(x)=B^{(0)}(x)+\sqrt{1-x^2}B^{(1)}(x)\), where \(B^{(0)}\in\mathbb P_m\) and \(B^{(1)}\in\mathbb P_{m-1}\) satisfy \(B^{(0)}(-x)=(-1)^mB^{(0)}(x)\) and \(B^{(1)}(-x)=(-1)^{m-1}B^{(1)}(x)\). The specific aim of the paper is to consider some properties, in particular, the orthogonal properties of the sequence of functions \(\{\mathcal W_m\}\), where \(\mathcal W_m\in\Omega_m\), given by NEWLINE\[NEWLINE\begin{aligned} \mathcal W_0(x)&=\gamma_0,\quad \mathcal W_1(x)=(\gamma_1x- \beta_1 \sqrt{1-x^2}) \gamma_0, \\ \mathcal W_{m+1}(x)&=[\gamma_{m+1}x-\beta_{m+1}\sqrt{1-x^2}]\mathcal W_m(x)-\alpha_{m+1}\mathcal W_{m-1}(x),\quad m\geq1. \end{aligned} NEWLINE\]NEWLINENEWLINENEWLINEHere \(\{\alpha_m\}_{m=2}^\infty\), \(\{\beta_m\}_{m=1}^\infty\) and \(\{\gamma_m\}_{m=0}^\infty\) are sequences of real numbers.NEWLINENEWLINEIn Section 2 a connection of functions in \(\Omega_m\) and self-inversive polynomials is investigated. For example, if \(\mathcal F\in\Omega_m\) then the number of zeros of \(\mathcal F\) in \((-1,1)\) cannot exceed \(m\) and moreover, if 1 is a zero of \(\mathcal F\), then \(-1\) is also a zero of \(\mathcal F\), and in this case the number of zeros of \(\mathcal F\) in \([-1,1]\) cannot exceed \(m +1\) (Theorem 2.2). Some basic properties of functions \(\mathcal F_m\) and \(\mathcal W_m\) are considered in Section 3.NEWLINENEWLINEOrthogonal properties associated with \(\mathcal W_m\) are investigated in Section 4. Quadrature rules based on the zeros of functions \(\mathcal W_m\) are given in Section 5. Here the interpolatory type quadrature rule at the zeros of \(\mathcal W_m\) is also considered. The connection with orthogonal polynomials on the unit circle is investigated in Section 6. Two examples are given in Section 7.