Skip to main content
SpringerLink
Log in
Book cover

AIMS-Volkswagen Stiftung Workshops

AIMSVSW 2018: Orthogonal Polynomials pp 215–244Cite as

Birkhäuser

Some Characterization Problems Related to Sheffer Polynomial Sets

  • Conference paper
  • First Online:

  • 958 Accesses

Abstract

In this work, we show some properties of Sheffer polynomials arising from quasi-monomiality. We survey characterization problems dealing with d-orthogonal polynomial sets of Sheffer type. We revisit some families in the literature and we state an explicit formula giving the exact number of Sheffer type d-orthogonal sets. We investigate, in detail, the (d + 1)-fold symmetric case as well as the particular cases d = 1, 2, 3.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. A.I. Aptekarev, Multiple polynomials. J. Comput. Appl. Math. 99, 423–447 (1998)

    Article  MathSciNet  Google Scholar 

  2. A.I. Aptekarev, V.A. Kaliaguine, W. Van Assche, Criterion for the resolvent set of non-symmetric tridiagonal operators. Proc. Amer. Math. Soc. 123, 2423–2430 (1995)

    Article  MathSciNet  Google Scholar 

  3. Y. Ben Cheikh, On obtaining dual sequences via quasi-monomiality. Georgian Math. J. 9, 413–422 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Y. Ben Cheikh, Some results on quasi-monomiality. Appl. Math. Comput. 141, 63–76 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Y. Ben Cheikh, N. Ben Romdhane, d-symmetric d-orthogonal polynomials of Brenke type. J. Math. Anal. Appl. 416, 735–747 (2014)

    Google Scholar 

  6. Y. Ben Cheikh, M. Gaied, Dunkl-Appell d-orthogonal polynomials. Integral Transform. Spec. Funct. 28, 581–597 (2007)

    Article  MathSciNet  Google Scholar 

  7. Y. Ben Cheikh, I. Gam, On some operators varying the dimensional parameters of d-orthogonality. Integral Transform. Spec. Funct. 27, 731–746 (2016)

    Article  MathSciNet  Google Scholar 

  8. Y. Ben Cheikh, A. Ouni, Some generalized hypergeometric d-orthogonal polynomial sets. J. Math. Anal. Appl. 343, 464–478 (2008)

    Article  MathSciNet  Google Scholar 

  9. Y. Ben Cheikh, A. Zaghouani, Some discrete d-orthogonal polynomial sets. J. Comput. Appl. Math. 156, 253–263 (2003)

    Article  MathSciNet  Google Scholar 

  10. Y. Ben Cheikh, A. Zaghouani, d-orthogonality via generating functions. J. Comput. Appl. Math. 199, 2–22 (2007)

    Article  MathSciNet  Google Scholar 

  11. A. Boukhemis, On the classical 2-orthogonal polynomials sequences of Sheffer-Meixner type. CUBO A Math. J. Univ. Frontera 7, 39–55 (2005)

    MathSciNet  MATH  Google Scholar 

  12. A. Boukhemis, P. Maroni, Une caractérisation des polynômes strictement 1∕p orthogonaux de type Sheffer. Étude du cas p = 2. J. Math. Approx. Theory. 54, 67–91 (1998)

    Google Scholar 

  13. E. Bourreau, Polynômes orthogonaux simultanés et systèmes dynamiques infinis. Ph.D. Thesis Université des Sciences et Technologie de Lille-Lille I. France (2002)

    Google Scholar 

  14. M.G. Bruin, Simultaneous Padé approximation and orthogonality, in Polynômes Orthogonaux et Applications, ed. by C. Brezinski, A. Draux, A.P. Magnus, P. Maroni, A. Ronveaux. Lecture Notes in Mathematics, vol. 1171 (Springer, Berlin, 1985), pp. 74–83

    Google Scholar 

  15. H. Chaggara, Quasi monomiality and linearization cofficients for Sheffer polynomial sets, in Proceedings of the International Conference: Difference Equations, Special Functions and Orthogonal Polynomials Munich, Germany, 2005. ed. by S. Elaydi, J. Cushing, R. Lasser, V. Papageorgiou, A. Ruffing, W. Van Assche. (Special edition of World Scientific, Singapore, 2007), pp. 90–99

    Google Scholar 

  16. H. Chaggara, R. Mbarki, On d-orthogonal polynomials of Sheffer type. J. Diff. Equ. Appl. 24, 1808–1829 (2018)

    Article  MathSciNet  Google Scholar 

  17. E. Coussement, W. Van Assche, Multiple orthogonal polynomials associated with the modified Bessel functions of the first kind. Constr. Approx. 19, 237–263 (2003)

    Article  MathSciNet  Google Scholar 

  18. K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials. J. Comput. Appl. Math. 70, 279–295 (1996)

    Article  MathSciNet  Google Scholar 

  19. K. Douak, On 2-orthogonal polynomials of Laguerre type. Inter. J. Math. Math. Sci. 22, 29–48 (1999)

    Article  MathSciNet  Google Scholar 

  20. K. Douak, P. Maroni, Les polynômes orthogonaux classiques de dimension deux. Analysis 12, 71–107 (1992)

    Article  MathSciNet  Google Scholar 

  21. H.W. Gould, A.T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29, 51–63 (1962)

    Article  MathSciNet  Google Scholar 

  22. M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable (Cambridge University Press, Cambridge, 2005)

    Book  Google Scholar 

  23. I. Kubo, Generating functions of exponential type for orthogonal polynomials. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 155–159 (2004)

    Article  MathSciNet  Google Scholar 

  24. P. Maroni, L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux. Ann. Fac. Sci. Toulouse Math. 10, 105–139 (1998)

    Article  Google Scholar 

  25. J. Meixner, Orthogonale polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. Lond. Math. Soc. 1, 6–13 (1934)

    Article  MathSciNet  Google Scholar 

  26. E.D. Rainville, Special Functions (The Macmillan Company, New York, 1960)

    MATH  Google Scholar 

  27. S. Roman, G.C. Rota, The Umbral Calculus. Adv. Math. 27, 95–188 (1978)

    Article  Google Scholar 

  28. A. Saib, On semi-classical d-orthogonal polynomials. Math. Nachr. 286, 1863–1885 (2013)

    Article  MathSciNet  Google Scholar 

  29. I.M. Sheffer, Some properties of polynomial sets of type zero. Duke Math. J. 5, 590–622 (1939)

    Article  MathSciNet  Google Scholar 

  30. J. Van Iseghem, Approximants de Padé vectoriels. Ph.D. Thesis, Université des Sciences et Techniques de Lille-Flandre-Artois (1987)

    Google Scholar 

  31. J. Van Iseghem, Vector orthogonal relations. Vector QD-algorithm. J. Comput. Appl. Math. 19, 141–150 (1987)

    Article  Google Scholar 

  32. S. Varma, Some new d-orthogonal polynomial sets of Sheffer type. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68, 1–10 (2018)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

We are much indebted to the organisers, Wolfram Koepf and Mama Foupouagnigni, of the AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications, Douala, Cameroon, October 5–12, 2018 for the invitation, the interesting program and the kind hospitality.

The first author would like to extend his appreciation to the Deanship of Scientific Research at King Khalid University for funding his work through research groups program under grant (2019).

Author information

Authors and Affiliations

  1. King Khalid University College of Sciences, Department of Mathematics, Abha, Kingdom of Saudi Arabia

    Hamza Chaggara

  2. École Supérieure des Sciences et de la Technologie, Université de Sousse, Sousse, Tunisia

    Hamza Chaggara & Salma Boussorra

  3. Institut Préparatoire aux Études d’Ingénieur de Monastir, Université de Monastir, Monastir, Tunisia

    Radhouan Mbarki

Authors
  1. Hamza Chaggara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radhouan Mbarki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Salma Boussorra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hamza Chaggara .

Editor information

Editors and Affiliations

  1. University of Yaoundé I, Yaoundé, Cameroon; African Institute for Mathematical Sciences, Limbe, Cameroon

    Mama Foupouagnigni

  2. Institute for Mathematics, University of Kassel, Kassel, Germany

    Wolfram Koepf

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chaggara, H., Mbarki, R., Boussorra, S. (2020). Some Characterization Problems Related to Sheffer Polynomial Sets. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_12

Download citation

Publish with us

Policies and ethics

Search

Navigation