A class of hypergeometric polynomials with zeros on the unit circle: extremal and orthogonal properties and quadrature formulas
The theory of special functions has been increasingly applied to the study of polynomials. Special hypergeometric functions are of great interest for the study of orthogonal polynomials on the real line. Consider the hypergeometric function \(_2F_1(a,b;c;z)\), commonly referred to as Gauss' hypergeometric function, i.e., \[ _2F_1 (a,b;c;z)=\sum_{n=0}^{\infty}{\frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{z^n}{n!}}, \] where \((\cdot)_{n}\) denotes the Pochhammer symbol and \(c\neq0,-1,-2,\dots\) In the cases where any of the parameters \(a\) and \(b\) are negative integers \(-m\), the series expansion terminates at \(n=m\), which reduces the function \(_2F_1(a,b;c;z)\) to a polynomial of degree \(m\). In the paper, the authors discuss some properties of two classes of hypergeometric polynomials, namely \[ R_{m}(b;z)=_2F_1 (-m,b;b+ \bar{b};1-z), \quad m\geq 0, \] \[ S_{m}(b;z)=_2F_1 (-m,b+1;b+ \bar{b}+1;1-z), \quad m\geq 0, \] where \(\operatorname{Re}(b)>0\). The class \(R_{m}(b;z)\) is of interest, since, when these polynomials are considered as functions of \(\operatorname{Im}(b)\), then they are Meixner-Pollaczek polynomials. The other class \(S_{m}(b;z)\), on the other hand, turns out to be interesting from the point of view of the theory of orthogonal polynomials on the unit circle. Also, the authors point out that \(\{R_{m}(b;z)\}\) is a special sequence of para-orthogonal polynomials associated with the sequence \(\{S_{m}(b;z)\}\) of orthogonal polynomials in the unit circle, or Szegő polynomials. The authors derive an interpolatory quadrature formula on the zeros of \(R_{m}(b;z)\). Also, related quadrature formulas for the interval \((-1,1)\) are derived which is based on the zeros of the auxiliary function \(G_{m}(\lambda,\eta;\chi)\). The latter can be defined as \[ G_{m}(\lambda,\eta;\chi)=\frac{(2\lambda)_{m}}{(\lambda)_{m}}(4z)^{-m/2}R_{m}(b;z), \quad m\geq 0, \] where \(\lambda\), \(\eta\) and \(\chi\) are such that \(b=\lambda +i\eta\) and \(2\chi=z^{1/2}+z^{-1/2}\). In order to achieve the objectives of the paper, at first, the authors present a few definitions concerning the orthogonal and para-orthogonal polynomials cited above. Among the definitions, the relation of the latter polynomials to their monic versions is presented. Following the preliminary results, the authors study the \(R_{m}(b;z)\) polynomials as kernel polynomials of the associated Szegő polynomials, \(S_{m}(b-1;z)\). The results concerning this topic are discussed under the light of the Christoffel-Darboux formula. Besides, some extremal properties for \(R_{m}(b;z)\) are discussed based on the properties of Gauss' hypergeometric functions, the minimal property of monic Szegő polynomials and the minimal and maximal properties of certain reproducing kernels. The same analysis performed for the \(R_{m}(b;z)\) polynomials is considered for the auxiliary function \(G_{m}(\lambda,\eta;\chi)\). Further, orthogonality issues concerning both the latter are discussed as the authors show that both the polynomials and the auxiliary function satisfy orthogonality relations. Finally, a quadrature formula based on the zeros of \(R_{m}(b;z)\) polynomials is obtained based on standard results from the literature. A modified approach allows the authors to provide formulas on the interval \((-1,1)\), which are related to the zeros of the auxiliary function \(G_{m}(\lambda,\eta;\chi)\).
scientific article; zbMATH DE number 6138080