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In this note we recast the Geronimus transformation in the framework of polynomials orthogonal with respect to symmetric bilinear forms. We also show that the double Geronimus transformations lead to non-diagonal Sobolev type inner products. ; The research of MD is supported by the European Research Council under the European Union Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259173. The research of FM has been supported by Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain, grant MTM2012-36732-C03-01. ; European Community's Seventh Framework Program

We study the interlacing properties of zeros of para-orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle d mu and para-orthogonal polynomials associated with a modification of d mu by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation. ; The authors thank the referees for their comments and suggestions. This work is partially supported by the CMUC, funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the Fundacao para a Ciência e a Tecnologia (FCT) under the project PEst-C/MAT/UI0324/2013. The research of the first author is supported by the Portuguese Government through the FCT under the grant SFRH/BPD/101139/2014. This author also acknowledges the financial support by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012–36732–C03–01. The second author also acknowledges the financial support by the Brazilian Government through the CAPES under the project 107/2012.

In this paper, we study new algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the recurrence coefficients, the so-called co-polynomials on the real line, are considered. We investigate the behavior of their zeros, mainly interlacing and monotonicity properties. Furthermore, using a transfer matrix approach we obtain new structural relations, combining theoretical and computational advantages. Finally, a connection with the theory of orthogonal polynomials on the unit circle is pointed out. ; The authors thank the referee for constructive comments and recommendations which improved the readability and quality of the manuscript. The research of the first author is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/ 101139/2014. This author also acknowledges the financial support by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012-36732-C03-01. The second author also acknowledges the financial support by the Brazilian Government through the CAPES under the project 107/2012.

A pair of regular linear functionals (U,V)(U,V) is said to be a (M,N)(M,N)-coherent pair of order (m,k)(m,k) if their corresponding sequences of monic orthogonal polynomials. ; We thank the referees for the careful revision of the manuscript. Their comments and suggestions have contributed to improve substantially the presentation. The work of F. Marcellán, J. Petronilho, and N.C. Pinzón-Cortés has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under grants MTM2012-36732-C03-01 (FM and NCP-C) and MTM2012-36732-C03-02 (JP). The work of J. Petronilho was also supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2011, and the research project PTDC/MAT/098060/2008 (FCT).

We present a new structure relation for the sequence of orthogonal polynomials associated with a D-v-semiclassical linear functional of class s, and then we use it to obtain a matrix characterization of the D-v-semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthonormal polynomials, and the nonsingular lower triangular matrix that represents the orthogonal polynomials with respect to some bases of polynomials. We also provide a matrix characterization of D-v-coherent pairs of linear functionals. ; The work of the first author was supported by a grant of the Secretar´ıa de Educaci´on P´ublica of M´exico and the Mexican Government. The work of the second author was supported by Consejo Nacional de Ciencia y Tecnolog´ıa of M´exico, grant 156668. The work of the third author was supported by Direcci ´on General de Investigaci´on Cient´ıfica y T´ecnica, Ministerio de Econom´ıa y Competitividad of Spain, grant MTM2012-36732-C03-01. The authors thank the anonymous referee for her/his valuable comments and suggestions. They contributed to improve the presentation of the manuscript.

We obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the Jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials, and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals. ; The work of the first author was supported by a grant of the Secretaría de Educación Pública of México and the Mexican Government. The work of the second author was supported by Consejo Nacional de Ciencia y Tecnología of México, Grant 156668. The work of the third author was supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, Grant MTM2012 36732 C03 01.