Skip to main content
Springer Nature Link
Log in

The L-Appell Symmetric Orthogonal Polynomials

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we shall be concerned with lowering operators defined on polynomials by means of

$$\begin{aligned} L(x^n)=\mu _nx^{n-1},\ \ n=0,1,\ldots , \ \mu _0=0,\ \ \mu _n\ne 0\ \ (n=1,2,\ldots ). \end{aligned}$$

We determine a necessary and sufficient condition on lowering operators L and a symmetric orthogonal polynomial sets \(\{P_n\}_{n\ge 0}\) such that \(\{P_n\}_{n\ge 0}\) is L-Appell. The resulting polynomials are the generalized Hermite and the symmetric PSs related to Wall and generalized Stieltjes–Wigert. Various properties of the obtained families are singled out: a three-term recurrence relation, explicit expression in term of hypergeometric and basic hypergeometric functions and generating functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Angelesco, A.: Sur les polynômes orthogonaux en rapport avec d’autres polynômes. Bul. Soc. Stiite Cluj. 1, 44–59 (1921)

    Google Scholar 

  2. Al-Salam, W.A.: \(q\)-Appell polynomials. Ann. Mat. Pura Appl. 77, 31–45 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Appell, P.: Sur une classe de polynômes. Ann. Sci. École Norm. Sup. 9, 119–144 (1880)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben Cheikh, Y., Gaied, M.: Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl. Math. Comput. 187, 105–114 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Ben Cheikh, Y., Gaied, M.: Dunkl-Appell \(d\)-orthogonal polynomials. Integral Transforms Spec. Func. 18, 581–597 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben Cheikh, Y., Gaied, M.: \(q\)-Dunkl-classical q-Hermite type polynomials. Georgian Math. J. 21, 125–137 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouanani, A., Khériji, L., Ihsen Tounsi, M.: Characterization of q-Dunkl Appell symmetric orthogonal q-polynomials. Expo. Math. 28, 325–336 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chihara, T.S.: Orthogonal polynomials with Brenke type generating functions. Duke Math. J. 35, 505–518 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordan and Breach, New York (1978)

    MATH  Google Scholar 

  10. Ghressi, A., Khériji, L.: On the q-analogue of Dunkl operator and its Appell classical orthogonal polynomials. Int. J. Pure Appl. Math. Sci. 39, 1–16 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and their q-analogues Springer Monographs in Mathematics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  12. Meixner, J.: Orthogonal polynomsysteme mit einer besondern gestalt der erzeugender functionen. J. Lond. Math. Soc. 9, 6–13 (1934)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Supérieur d’Informatique et des Techniques de Communication de Hammam Sousse, Sousse University, Sousse, Tunisia

    Mohamed Gaied

Authors
  1. Mohamed Gaied
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed Gaied.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gaied, M. The L-Appell Symmetric Orthogonal Polynomials. Mediterr. J. Math. 14, 211 (2017). https://doi.org/10.1007/s00009-017-1006-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-017-1006-7

Keywords

Mathematics Subject Classification

Search

Navigation