The Ramanujan Journal

, Volume 46, Issue 1, pp 29–48 | Cite as

Connection and inversion coefficients for basic hypergeometric polynomials

  • Hamza Chaggara
  • Mohamed Mabrouk


In this paper, we give a closed-form expression of the inversion and the connection coefficients for general basic hypergeometric polynomial sets using some known inverse relations. We derive expansion formulas corresponding to all the families within the q-Askey scheme and we connect some d-orthogonal basic hypergeometric polynomials.


Connection coefficients Inversion coefficients Basic hypergeometric polynomials Inverse relations q-Askey scheme d-orthogonal basic polynomials 

Mathematics Subject Classification

33C45 41A10 41A58 



Sincere thanks are due to the referee for his/her careful reading of the manuscript and for his/her valuable comments and suggestions which have considerably improved the quality of this paper.


  1. 1.
    Álvarez-Nodarse, R., Aversú, J., Yáñez, R.J.: On the connection and linearization problem for discrete hypergeometric \(q\)-polynomials. J. Math. Anal. Appl. 135, 52–78 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, G.E.: Connection coefficient problems and partitions. Proc. Symp. Pure Math. Am. Math. Soc. 34, 1–24 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Area, I., Godoy, E., Ronveaux, A., Zarzo, A.: Inversion problems in the \(q\)-Hahn tableau. J. Symb. Comput. 28, 767–776 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Area, I., Godoy, E., Ronveaux, A., Zarzo, A.: Solving connection and linearization problems within the Askey Scheme and its \(q\)-analogue via inversion formulas. J. Comput. Appl. Math. 133, 151–162 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Askey, R., Wilson, J.: A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols. SIAM J. Math. Anal. 8, 1008–1016 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Askey, R., Wilson, J.: Some Basic Hypergeometric Polynomials that Generalize Jacobi Polynomials. Memoirs of the American Mathematical Society, vol. 319. American Mathematical Society, Providence (1985)zbMATHGoogle Scholar
  7. 7.
    Ben Cheikh, Y., Chaggara, H.: Connection coefficients between Boas-Buck polynomial sets. J. Math. Anal. Appl. 319, 665–689 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ben Cheikh, Y., Lamiri, I., Ouni, A.: \(d\)-Orthogonality of little \(q\)-Laguerre type polynomials. J. Comput. Appl. Math. 236, 74–84 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ben Romdhane, N.: A general theorem on inversion problems for polynomial sets. Mediterr. J. Math. 13(5), 2783–2793 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ben Romdhane, N., Gaied, M.: A generalization of the symmetric classical polynomials: Hermite and Gegenbauer polynomials. Integral Transform. Spec. Funct. 27, 227–244 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carlitz, L.: Some inverse relations. Duke Math. J. 40, 893–901 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Foupouagnigni, M., Koepf, W., Tcheutia, D.D.: Connection and linearization coefficients for the Askey-Wilson polynomials. J. Symb. Comput. 53, 96–118 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Foupouagnigni, M., Koepf, W., Tcheutia, D.D., Sadjang, N.N.: Representation of \(q\)-orthogonal polynomials. J. Symb. Comput. 47, 1347–1371 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gasper, G.: Projection formulas for orthogonal polynomials of a discrete variable. J. Math. Anal. Appl. 45, 176–198 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications, vol. 35. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  16. 16.
    Gould, H.W., Hsu, : Some new inverse series relations. Duke Math. J 40, 885861 (1973)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Koekoek, R., Lesky, P.A., Swarrtow, R.F.: Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer, Berlin (2010)CrossRefGoogle Scholar
  19. 19.
    Krattenthaler, C.: A new matrix inverse. Proc. Am. Math. Soc. 124, 47–59 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lamiri, I., Ouni, A.: \(d\)-Orthogonality of some basic hypergeometric polynomials. Georgian Math. J. 20, 729–751 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Maroni, P.: L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux. Ann. Fac. Sci. Toulouse 10, 105–139 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Maroni, P., da Rocha, Z.: Connection coefficients for orthogonal polynomials: symbolic computations, verifications and demonstrations in the Mathematica language. Numer. Algor. 63, 507–520 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Njinou Sadjang, P., Foupouagnigni, M., Koepf, W.: On moments of classical orthogonal polynomials. J. Math. Anal. Appl. 424, 122–151 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)zbMATHGoogle Scholar
  25. 25.
    Riordan, J.: Combinatorial Identities. Wiley, New York (1968)zbMATHGoogle Scholar
  26. 26.
    Stokman, J.V., Koornwinder, T.H.: On some limit cases of Askey-Wilson polynomials. J. Approx. Theory 95, 310–330 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Higher School of Sciences and TechnologySousse UniversitySousseTunisia

Personalised recommendations