Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials

  • Daniel Duviol TcheutiaEmail author
Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


Our main objective is to establish the so-called connection formula,
$$\displaystyle \begin{aligned} p_n(x)=\sum_{k=0}^{n}C_{k}(n)y_k(x), \end{aligned} $$
which for pn(x) = xn is known as the inversion formula
$$\displaystyle \begin{aligned} x^n=\sum _{k=0}^{n}I_{k}(n)y_k(x), \end{aligned} $$
for the family yk(x), where \(\{p_n(x)\}_{n\in \mathbb {N}_0}\) and \(\{y_n(x)\}_{n\in \mathbb {N}_0}\) are two polynomial systems. If we substitute x by ax in the left hand side of (0.1) and yk by pk, we get the multiplication formula
$$\displaystyle \begin{aligned} p_n(ax)=\sum _{k=0}^{n}D_{k}(n,a)p_k(x). \end{aligned} $$
The coefficients Ck(n), Ik(n) and Dk(n, a) exist and are unique since deg pn = n, deg yk = k and the polynomials {pk(x), k = 0, 1, …, n} or {yk(x), k = 0, 1, …, n} are therefore linearly independent. In this session, we show how to use generating functions or the structure relations to compute the coefficients Ck(n), Ik(n) and Dk(n, a) for classical continuous orthogonal polynomials.


Orthogonal polynomials Inversion coefficients Multiplication coefficients Connection coefficients 

Mathematics Subject Classification (2000)

33C45 33D45 33D15 33F10 68W30 



Many thanks to the organizers of the AIMS–Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications, Douala, October 5–12, 2018.


  1. 1.
    I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Solving connection and linearization problems within the Askey scheme and its q-analogue via inversion formulas. J. Comput. Appl. Math. 136, 152–162 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    R. Askey, G. Gasper, Jacobi polynomial expansions of Jacobi polynomials with nonnegative coefficients. Proc. Camb. Philos. Soc. 70, 243–255 (1971)CrossRefGoogle Scholar
  3. 3.
    R. Askey, G. Gasper, Convolution structures for Laguerre polynomials. J. Anal. Math. 31, 48–68 (1977)MathSciNetCrossRefGoogle Scholar
  4. 4.
    H. Chaggara, W. Koepf, Duplication coefficients via generating functions. Complex Var. Elliptic Equ. 52, 537–549 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    T. Cluzeau, M. van Hoeij, Computing hypergeometric solutions of linear recurrence equations. Appl. Algebra Eng. Commun. Comput. 17, 83–115 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    E.H. Doha, H.M. Ahmed, Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials. J. Phys. A 37, 8045–8063 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.L. Fields, J. Wimp, Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15, 390–395 (1961)MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Godoy, A. Ronveaux, A. Zarzo, I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case. J. Comput. Appl. Math. 84, 257–275 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)Google Scholar
  10. 10.
    W. Koepf, Hypergeometric Summation—An Algorithmic Approach to Summation and Special Function Identities, 2nd edn. (Springer Universitext, Springer, London, 2014)CrossRefGoogle Scholar
  11. 11.
    W. Koepf, D. Schmersau, Representations of orthogonal polynomials. J. Comput. Appl. Math. 90, 57–94 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Lewanowicz, The hypergeometric functions approach to the connection problem for the classical orthogonal polynomials. Technical Report, Institute of Computer Science, University of Wroclaw (2003)zbMATHGoogle Scholar
  13. 13.
    P. Njionou Sadjang, Moments of classical orthogonal polynomials, Ph.D. Dissertation, Universität Kassel (2013)zbMATHGoogle Scholar
  14. 14.
    M. Petkovšek, Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comput. 14, 243–264 (1992)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Petkovšek, H. Wilf, D. Zeilberger, A = B (A. K. Peters, Wellesley, 1996)Google Scholar
  16. 16.
    E.D. Rainville, Special Functions (The Macmillan Company, New York, 1960)zbMATHGoogle Scholar
  17. 17.
    A. Ronveaux, A. Zarzo, E. Godoy, Recurrence relations for connection between two families of orthogonal polynomials. J. Comput. Appl. Math. 62, 67–73 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Sánchez-Ruiz, J.S. Dehesa, Expansions in series of orthogonal hypergeometric polynomials. J. Comput. Appl. Math. 89, 155–170 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    D.D. Tcheutia, On connection, linearization and duplication coefficients of classical orthogonal polynomials, Ph.D. Dissertation, Universität Kassel (2014)Google Scholar
  20. 20.
    D.D. Tcheutia, M. Foupouagnigni, W. Koepf, P. Njionou Sadjang, Coefficients of multiplication formulas for classical orthogonal polynomials. Ramanujan J. 39, 497–531 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. van Hoeij, Finite singularities and hypergeometric solutions of linear recurrence equations. J. Pure Appl. Algebra 139, 109–131 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Zarzo, I. Area, E. Godoy, A. Ronveaux, Results for some inversion problems for classical continuous and discrete orthogonal polynomials. J. Phys. A 30, 35–40 (1997)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of KasselKasselGermany

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