Second-order difference equation for Sobolev-type orthogonal polynomials: Part I: theoretical results

We consider a general Sobolev-type inner product involving the Hahn difference operator, so this includes the well-known difference operators and Δ and, as a limit case, the derivative operator. The objective is to construct the ladder operators for the corresponding nonstandard orthogonal polynomials and deduce the second-order differential-difference equation satisfied by these polynomials. Moreover, we will show that all the functions involved in these constructions can be computed explicitly.

[1]  J. Petronilho,et al.  On discrete coherent pairs of measures , 2020, Journal of Difference Equations and Applications.

[2]  F. Marcell'an,et al.  $$H_q$$ H q -Semiclassical orthogonal polynomials via polynomial mappings , 2016, The Ramanujan Journal.

[3]  J. Petronilho,et al.  On classical orthogonal polynomials related to Hahn's operator , 2019, Integral Transforms and Special Functions.

[4]  Edmundo J. Huertas,et al.  On Freud–Sobolev type orthogonal polynomials , 2017, Afrika Matematika.

[5]  Juan F. Mañas-Mañas,et al.  Ladder operators and a differential equation for varying generalized Freud-type orthogonal polynomials , 2018, Random Matrices: Theory and Applications.

[6]  R. S. Costas-Santos,et al.  Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials , 2018, Journal of Difference Equations and Applications.

[7]  Edmundo J. Huertas,et al.  New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials , 2017, Numerical Algorithms.

[8]  W. Assche.  Orthogonal Polynomials and Painlevé Equations , 2017 .

[9]  Francisco Marcellán,et al.  Asymptotics for varying discrete Sobolev orthogonal polynomials , 2017, Appl. Math. Comput..

[10]  I. Sharapudinov,et al.  Sobolev Orthogonal Polynomials Generated by Meixner Polynomials , 2016 .

[11]  Francisco Marcellán,et al.  On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight , 2015, Appl. Math. Comput..

[12]  Juan J. Moreno-Balcázar,et al.  Δ-Meixner-Sobolev orthogonal polynomials: Mehler-Heine type formula and zeros , 2015, J. Comput. Appl. Math..

[13]  J. Petronilho,et al.  On linearly related sequences of difference derivatives of discrete orthogonal polynomials , 2014, J. Comput. Appl. Math..

[14]  J. Petronilho,et al.  Variations around Jackson’s quantum operator , 2015 .

[15]  Yuan Xu,et al.  On Sobolev orthogonal polynomials , 2014, 1403.6249.

[16]  Mahmoud H. Annaby,et al.  Hahn Difference Operator and Associated Jackson–Nörlund Integrals , 2012, J. Optim. Theory Appl..

[17]  Juan J. Moreno-Balcázar,et al.  The semiclassical Sobolev orthogonal polynomials: A general approach , 2010, J. Approx. Theory.

[18]  Rene F. Swarttouw,et al.  Hypergeometric Orthogonal Polynomials , 2010 .

[19]  Yang Chen,et al.  Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I , 2008, J. Approx. Theory.

[20]  M. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable: Bibliography , 2005 .

[21]  Yang Chen,et al.  Ladder Operators for q-orthogonal Polynomials , 2007, 0711.2454.

[22]  M. Schlosser.  BASIC HYPERGEOMETRIC SERIES , 2007 .

[23]  Rene F. Swarttouw,et al.  Orthogonal Polynomials , 2005, Series and Products in the Development of Mathematics.

[24]  Yang Chen,et al.  Orthogonal polynomials with discontinuous weights , 2005, math-ph/0501057.

[25]  I. Area,et al.  \Delta -Coherent Pairs and Orthogonal Polynomials of a Discrete Variable , 2003 .

[26]  M. Anshelevich,et al.  Introduction to orthogonal polynomials , 2003 .

[27]  Francisco Marcellán,et al.  q-Coherent pairs and q-orthogonal polynomials , 2002, Appl. Math. Comput..

[28]  Yang Chen,et al.  Krall-type polynomials via the Heine formula , 2002 .

[29]  I. Area,et al.  Inner products involving q -differences: the little q Laguerre-Sobolev polynomials , 2000 .

[30]  R. Koekoek,et al.  Difference operators with Sobolev type meixner polynomials as eigenfunctions , 1998 .

[31]  Yang Chen,et al.  Ladder operators and differential equations for orthogonal polynomials , 1997 .

[32]  Francisco Marcellán,et al.  On orthogonal polynomials of Sobolev type: algebraic properties and zeros , 1992 .

[33]  R. Koekoek.  Generalizations of a q -analogue of Laguerre polynomials , 1992 .

[34]  R. Koekoek.  A Generalization of Moak's q-Laguerre Polynomials , 1990, Canadian Journal of Mathematics.

[35]  Francisco Marcellán,et al.  On a class of polynomials orthogonal with respect to a discrete Sobolev inner product , 1990 .

[36]  W. Hahn.  Über Orthogonalpolynome, die q-Differenzengleichungen genügen , 1949 .

[37]  J. C. Tressler,et al.  Fourth Edition , 2006 .