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The efficacy of numerical methods like integral estimates via Gaussian quadrature formulas depends on the localization of the zeros of the associated family of orthogonal polynomials. In this regard, following the renewed interest in quadrature formulas on the unit circle, and $R_{II}$-type polynomials, which include the complementary Romanovski-Routh polynomials, in this work we present a collection of properties of their zeros. Our results include extreme bounds, convexity, and density, alongside the connection of such polynomials to classical orthogonal polynomials via asymptotic formulas.

We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special $R_{II}$ recurrence relation. We also look into some methods for generating the nodes (which lie on the real line) and the positive weights of these quadrature rules. With a simple transformation these quadrature rules on the real line also lead to certain positive quadrature rules of highest algebraic degree of precision on the unit circle. This way, we also introduce new approaches to evaluate the nodes and weights of these specific quadrature rules on the unit circle., Comment: 32 pages

We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follow from the Romanovski-Routh polynomials or complexified Jacobi polynomials, are known to be useful objects in the studies of the one-dimensional Schr\"{o}dinger equation and also the wave functions of quarks. One of the main results of this paper is to show how the CRR-polynomials are related to a special class of orthogonal polynomials on the unit circle. As another main result, we have established their connection to a class of functions which are related to a subfamily of Whittaker functions that includes those associated with the Bessel functions and the regular Coulomb wave functions. An electrostatic interpretation for the zeros of CRR-polynomials is also considered.

Given a nontrivial positive measure $\mu$ on the unit circle, the associated Christoffel-Darboux kernels are $K_n(z, w;\mu) = \sum_{k=0}^{n}\overline{\varphi_{k}(w;\mu)}\,\varphi_{k}(z;\mu)$, $n \geq 0$, where $\varphi_{k}(\cdot; \mu)$ are the orthonormal polynomials with respect to the measure $\mu$. Let the positive measure $\nu$ on the unit circle be given by $d \nu(z) = |G_{2m}(z)|\, d \mu(z)$, where $G_{2m}$ is a conjugate reciprocal polynomial of exact degree $2m$. We establish a determinantal formula expressing $\{K_n(z,w;\nu)\}_{n \geq 0}$ directly in terms of $\{K_n(z,w;\mu)\}_{n \geq 0}$. Furthermore, we consider the special case of $w=1$; it is known that appropriately normalized polynomials $K_n(z,1;\mu) $ satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters $ \{c_n(\mu)\}_{n=1}^{\infty}$ and $ \{g_{n}(\mu)\}_{n=1}^{\infty}$, with $0

When a measure $\psi(x)$ on the real line is subjected to the modification $d\psi^{(t)}(x) = e^{-tx} d \psi(x)$, then the coefficients of the recurrence relation of the orthogonal polynomials in $x$ with respect to the measure $\psi^{(t)}(x)$ are known to satisfy the so-called Toda lattice formulas as functions of $t$. In this paper we consider a modification of the form $e^{-t(p x+q/x)}$ of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either $p =0$ or $q=0$. However, as far as Lax pair representation is concern, no complete Lax pair representations were established for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established

The sequence $\{\,_2\phi_1(q^{-k},q^{b+1};\,q^{-\overline{b}-k+1};\, q, q^{-\overline{b}+1/2} z)\}_{k \geq 0}$ of basic hypergeometric polynomials is known to be orthogonal on the unit circle with respect to the weight function $|(q^{1/2}e^{i\theta};\,q)_{\infty}/(q^{b+1/2}e^{i\theta};\,q)_{\infty}|^2$. This result, where one must take the parameters $q$ and $b$ to be $0 < q < 1$ and $\Re(b) > -1/2$, is due to P.I. Pastro \cite{Pastro-1985}. In the present manuscript we deal with the orthogonal polynomials $\hat{\Phi}_{n}(b;.)$ and $\check{\Phi}_{n}(b;.)$ on the unit circle with respect to the two parametric families of weight functions $\hat{\omega}(b; \theta) = |(e^{i\theta};\,q)_{\infty}/(q^{b}e^{i\theta};\,q)_{\infty}|^2$ and $\check{\omega}(b;\theta) = |(qe^{i\theta};\,q)_{\infty}/(q^{b}e^{i\theta};\,q)_{\infty}|^2$, where $0 < q < 1$ and $\Re(b) > 0$. With the use of the basic hypergeometric polynomials $ _2\phi_1(q^{-k},q^{b};\,q^{-\overline{b}-k+1};\, q, q^{-\overline{b}+1} z)$, $k \geq 0$, which have zeros on the unit circle when $\Re(b) > 0$, simple expressions for the (monic) polynomials $\hat{\Phi}_{n}(b;.)$ and $\check{\Phi}_{n}(b;.)$, their norms, the associated Verblunsky coefficients and also the respective Szeg\H{o} functions are found.

It was shown recently that associated with a pair of real sequences $\{\{c_{n}\}_{n=1}^{\infty}, \{d_{n}\}_{n=1}^{\infty}\}$, with $\{d_{n}\}_{n=1}^{\infty}$ a positive chain sequence, there exists a unique nontrivial probability measure $\mu$ on the unit circle. The Verblunsky coefficients $\{\alpha_{n}\}_{n=0}^{\infty}$ associated with the orthogonal polynomials with respect to $\mu$ are given by the relation $$ \alpha_{n-1}=\overline{\tau}_{n-1}\left[\frac{1-2m_{n}-ic_{n}}{1-ic_{n}}\right], \quad n \geq 1, $$ where $\tau_0 = 1$, $\tau_{n}=\prod_{k=1}^{n}(1-ic_{k})/(1+ic_{k})$, $n \geq 1$ and $\{m_{n}\}_{n=0}^{\infty}$ is the minimal parameter sequence of $\{d_{n}\}_{n=1}^{\infty}$. In this manuscript we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences $\{c_{n}\}_{n=1}^{\infty}$ and $\{m_{n}\}_{n=1}^{\infty}$. When the sequence $ \{c_{n}\}_{n=1}^{\infty}$ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of $z= -1$. Furthermore, we show that it is possible to ge\-nerate periodic Verblunsky coefficients by choosing periodic sequences $\{c_{n}\}_{n=1}^{\infty}$ and $\{m_{n}\}_{n=1}^{\infty}$ with the additional restriction $c_{2n}=-c_{2n-1}, \, n\geq 1.$ We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained.

We consider a sequence of polynomials $\{P_n\}_{n \geq 0}$ satisfying a special $R_{II}$ type recurrence relation where the zeros of $P_n$ are simple and lie on the real line. It turns out that the polynomial $P_n$, for any $n \geq 2$, is the characteristic polynomial of a simple $n \times n$ generalized eigenvalue problem. It is shown that with this $R_{II}$ type recurrence relation one can always associate a positive measure on the unit circle. The orthogonality property satisfied by $P_n$ with respect to this measure is also obtained. Finally, examples are given to justify the results.

Given a non-trivial Borel measure $\mu$ on the unit circle $\mathbb T$, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at $z=1$ constitute a family of so-called para-orthogonal polynomials, whose zeros belong to $\mathbb T$. With a proper normalization they satisfy a three-term recurrence relation determined by two sequence of real coefficients, $\{c_n\}$ and $\{d_n\}$, where $\{d_n\}$ is additionally a positive chain sequence. Coefficients $(c_n,d_n)$ provide a parametrization of a family of measures related to $\mu$ by addition of a mass point at $z=1$. In this paper we estimate the location of the extreme zeros (those closest to $z=1$) of the para-orthogonal polynomials from the $(c_n,d_n)$-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of $\mu$ at $z=1$. These results are easily reformulated in order to find gaps in the support of $\mu$ at any other $z\in \mathbb T$. We provide also some examples showing that the bounds are tight and illustrating their computational applications.

When a nontrivial measure $\mu$ on the unit circle satisfies the symmetry $d\mu(e^{i(2\pi-\theta)}) = - d\mu(e^{i\theta})$ then the associated OPUC, say $S_n$, are all real. In this case, Delsarte and Genin, in 1986, have shown that the two sequences of para-orthogonal polynomials $\{zS_{n}(z) + S_{n}^{\ast}(z)\}$ and $\{zS_{n}(z) - S_{n}^{\ast}(z)\}$ satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval $[-1,1]$. The same authors, in (1988), have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently in Costa, Felix and Sri Ranga (2013) and then in Castillo, Costa, Sri Ranga and Veronese (2014), the extension associated with the para-orthogonal polynomials $zS_{n}(z) - S_{n}^{\ast}(z)$ was thoroughly explored, especially from the point of view of the three term recurrence, and chain sequences play an important part in this exploration. The main objective of the present manuscript is to provide the theory surrounding the extension associated with the para-orthogonal polynomials $zS_{n}(z) + S_{n}^{\ast}(z)$ for any nontrivial measure on the unit circle. Like in Costa, Felix and Sri Ranga (2013) and Castillo, Costa, Sri Ranga and Veronese (2014), chain sequences also play an important role in this theory. Examples and applications are also provided to justify the results obtained.

The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formula % \[ R_{n+1}(z) = \big[(1+ic_{n+1})z+(1-ic_{n+1})\big] R_{n}(z) - 4 d_{n+1} z R_{n-1}(z), \quad n \geq 1, \] % with $R_{0}(z) =1$ and $R_{1}(z) = (1+ic_{1})z+(1-ic_{1})$, where $\{c_n\}_{n=1}^{\infty}$ is a real sequence and $\{d_n\}_{n=1}^{\infty}$ is a positive chain sequence. We establish that there exists an unique nontrivial probability measure $\mu$ on the unit circle for which $\{R_n(z) - 2(1-m_n)R_{n-1}(z)\}$ gives the sequence of orthogonal polynomials. Here, $\{m_n\}_{n=0}^{\infty}$ is the minimal parameter sequence of the positive chain sequence $\{d_n\}_{n=1}^{\infty}$. The element $d_1$ of the chain sequence, which does not effect the polynomials $R_n$, has an influence in the derived probability measure $\mu$ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if $\{M_n\}_{n=0}^{\infty}$ is the maximal parameter sequence of the chain sequence, then the measure $\mu$ is such that $M_0$ is the size of its mass at $z=1$. An example is also provided to completely illustrates the results obtained., Comment: 15 pages, 1 figure

The main goal in this manuscript is to present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to the class of symmetric orthogonal polynomials on $[-1,1]$, has a complete connection to the orthogonal polynomials on the unit circle. Quadrature rules and other properties based on the zeros of these functions are also considered., Comment: 20 pages, 2 figures