Two variable Freud orthogonal polynomials and matrix Painlevé-type difference equations

We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyse relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the coefficients of the structure relations satisfied by these bivariate semiclassical orthogonal polynomials, also a matrix differentialdifference equation for the bivariate orthogonal polynomials is deduced. The extension of the Painlev´e equation for the coefficients of the three term relations to the bivariate case and a two dimensional version of the Langmuir lattice are obtained.Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES) 88887.310463/2018-00 88887.575407/2020-00FEDER/Junta de Andalucia A-FQM-246-UGR20MCIN PGC2018-094932B-I00European CommissionIMAG-Maria de Maeztu grant CEX2020-00 1105-

On multivariate orthogonal polynomials and elementary symmetric functions

Acknowledgements The authors would like to express their gratitude to the two anonymous reviewers for their useful comments and suggestions, which improved the comprehension of the manuscript. In particular, we thank the reviewer who pointed out references [4–6, 15].Funding for open access charge: Universidad de Granada / CBUA This research was supported through the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES), in the scope of the CAPES-PrInt Program, process number 88887.310463/2018-00, International Cooperation Project number 88887.468471/2019-00. The second author (MAP) has been partially supported by grant PGC2018-094932-B-I00 from FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación, and the IMAG-María de Maeztu grant CEX2020-001105-M/ AEI/10.13039/501100011033.We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi−xj|2γ+1,x∈(a,b)d, for γ>−1 , where ω(t) is an univariate weight function in t∈(a,b) and x=(x1,x2,…,xd) with xi∈(a,b). Applying the change of variables xi, i=1,2,…,d, into ur, r=1,2,…,d, where ur is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having xi, i=1,2,…,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables ur for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables.Funding for open access charge: Universidad de Granada / CBUABrazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES), in the scope of the CAPES-PrInt Program, process number 88887.310463/2018-00, International Cooperation Project number 88887.468471/2019-00Grant PGC2018-094932-B-I00 from FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de InvestigaciónIMAG-María de Maeztu grant CEX2020-001105-M/ AEI/10.13039/50110001103

Mehler-Heine asymptotics of a class of generalized hypergeometric polynomials

We obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures

Quasi-analytical root-finding for non-polynomial functions

Made available in DSpace on 2018-12-11T16:46:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2017-11-01Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)A method is presented for the calculation of roots of non-polynomial functions, motivated by the requirement to generate quadrature rules based on non-polynomial orthogonal functions. The approach uses a combination of local Taylor expansions and Sturm’s theorem for roots of a polynomial which together give a means of efficiently generating estimates of zeros which can be polished using Newton’s method. The technique is tested on a number of realistic problems including some chosen to be highly oscillatory and to have large variations in amplitude, both of which features pose particular challenges to root–finding methods.Departamento de Matemática Aplicada UNESP–University Estadual PaulistaDepartment of Mechanical Engineering University of BathDepartamento de Matemática Aplicada UNESP–University Estadual PaulistaFAPESP: 2014/17357-1FAPESP: 2014/22571-

On semi-classical weight functions on the unit circle

We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at most 2. We determine all such semi-classical weight functions and this also includes an extension of the Jacobi weight function on the unit circle. General structure relations for the orthogonal polynomials and non-linear difference equations for the associated complex Verblunsky coefficients are established. As application, we present several new structure relations and non-linear difference equations associated with some of these semi-classical weight functions.Comment: 26 page

Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

AbstractWe consider the Sobolev inner product 〈f,g〉=∫−11f(x)g(x)(1−x2)α−12dx+∫f′(x)g′(x)dψ(x),α>−12, where dψ is a measure involving a Gegenbauer weight and with mass points outside the interval (−1,1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product. We obtain the asymptotics of the largest zeros of these polynomials via a Mehler–Heine type formula. These results are illustrated with some numerical experiments