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Classical Continuous Orthogonal Polynomials

  • Salifou MboutngamEmail author
Conference paper
  • 40 Downloads
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

Abstract

Classical orthogonal polynomials (Hermite, Laguerre, Jacobi and Bessel) constitute the most important families of orthogonal polynomials. They appear in mathematical physics when Sturn-Liouville problems for hypergeometric differential equation are studied. These families of orthogonal polynomials have specific properties. Our main aim is to:
  1. 1.

    recall the definition of classical continuous orthogonal polynomials;

     
  2. 2.

    prove the orthogonality of the sequence of the derivatives;

     
  3. 3.

    prove that each element of the classical orthogonal polynomial sequence satisfies a second-order linear homogeneous differential equation;

     
  4. 4.

    give the Rodrigues formula.

     

Keywords

Classical orthogonal polynomials Rodrigues formula Differential equation Pearson type equation 

Mathematics Subject Classification (2000)

33C45 33D45 

References

  1. 1.
    W.A. Al-Salam, Characterization theorems for orthogonal polynomials, in Orthogonal Polynomials: Theory and Practice, ed. by P. Nevai. NATO ASI Series C: Mathematical and Physical Sciences, vol. 294 (Kluwer, Dordrecht, 1990), pp. 1–24Google Scholar
  2. 2.
    W.A. Al-Salam, T.S. Chihara, Another characterization of the classical orthogonal polynomials. SIAM J. Math. Anal. 3, 65–70 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    T.S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978), p. 259zbMATHGoogle Scholar
  4. 4.
    M. Foupouagnigni, On difference equations for orthogonal polynomials on nonuniform lattices. J. Diff. Eqn. Appl. 14, 127–174 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A.G. García, F. Marcellán, L. Salto, A distributional study of discrete classical orthogonal polynomials. J. Comput. Appl. Math. 5(7), 147–162 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, in Springer Monographs in Mathematics (Springer, Berlin, 2010), p. 578CrossRefGoogle Scholar
  7. 7.
    F. Marcellán, A. Branquinho, J. Petronilho, Classical orthogonal polynomials: a functional approach. Acta Appl. Math. 34, 283–303 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Higher Teachers’ Training CollegeThe University of MarouaMarouaCameroon

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