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Polynomial sequences associated with the classical linear functionals

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Abstract

This work in mainly devoted to the study of polynomial sequences, not necessarily orthogonal, defined by integral powers of certain first order differential operators in deep connection to the classical polynomials of Hermite, Laguerre, Bessel and Jacobi. This connection is streamed from the canonical element of their dual sequences. Meanwhile new Rodrigues-type formulas for the Hermite and Bessel polynomials are achieved.

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References

  1. Carlitz, L.: On arrays of numbers. Amer. J. Math. 54, 739–752 (1932)

    Article  MathSciNet  Google Scholar 

  2. Comtet, L.: Advanced combinatorics—the art of finite and infinite expansions. Kluwer, Dordrecht (1974)

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

  4. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. III. McGraw-Hill, NewYork, London, Toronto (1953)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Hsu, L.C., Shiue, P.J-S.: A unified approach to generalized Stirling numbers. Adv. Appl. Math. 20, 366–384 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Janjić, M.: Some classes of numbers and derivatives. J. Integer Seq. 12, Article 09.8.3 (2009)

  8. Loureiro, A.F., Maroni, P., Yakubovich, S.: On a nonorthogonal polynomial sequence associated with Bessel operator. (arXiv:1104.4055v1). Proc. Amer. Math. Soc. (2012, to appear)

  9. Maroni, P.: L’orthogonalité et les récurrencesde polynômes d’ordre supérieur à deux. Ann. Fac. Sci. Toulouse 10, 1–36 (1989)

    Article  MathSciNet  Google Scholar 

  10. Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques. In: Brezinski, C., et al. (eds.) Orthogonal Polynomials and their Applications. IMACS Ann. Comput. Appl. Math., vol. 9, pp. 95–130 (1991)

  11. Maroni, P.: Variations around classical orthogonal polynomials. Connected problems. J. Comput. Appl. Math. 48, 133–155 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Maroni, P.: Fonctions Eulériennes. Polynômes orthogonaux classiques. Techniques de l’Ingénieur, traité Généralités (Sciences Fondamentales), A 154, 1–30 (1994)

  13. Maroni, P., Mejri, M.: Generalized Bernoulli polynomials revisited and some other Appell sequences. Georgian Math. J. 12(4), 697–716 (2005)

    MathSciNet  Google Scholar 

  14. OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences. http://oeis.org (2011)

  15. Sheffer, I.M.: Some properties of polynomial sets of type zero. Duke Math. J. 5(3), 590–622 (1939)

    Article  MathSciNet  Google Scholar 

  16. Toscano, L.: Numeri di Stirling generalizzati, operatori differenziali e polinomi ipergeometrici. Pont. Acad. Sci. Comment. 3, 721–757 (1939) (Italian)

    MathSciNet  Google Scholar 

  17. Toscano, L.: Sulla iterazione dell’operatore xD. Univ. Roma. Ist. Naz. Alta Mat. Rend. Mat. e Appl. 8(5), 337–350 (1949) (Italian)

    MathSciNet  MATH  Google Scholar 

  18. Van Iseghem, J.: Vector orthogonal relations. Vector QD-algorithm. J. Comput. Appl. Math. 19(1), 141–150 (1987)

    MATH  Google Scholar 

  19. Yakubovich, S.: A class of polynomials and discrete transformations associated with the Kontorovich–Lebedev operators. Integral Transforms Spec. Funct. 20, 551–567 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Ana Filipa Loureiro.

Additional information

Work of AFL supported by Fundação para a Ciência e Tecnologia via the grant SFRH/BPD/63114/2009. This research is partially funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese government through the FCT (Fundação para a Ciência e a Tecnologia) under the project PEst-C/MAT/UI0144/2011.

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Loureiro, A.F., Maroni, P. Polynomial sequences associated with the classical linear functionals. Numer Algor 60, 297–314 (2012). https://doi.org/10.1007/s11075-012-9573-y

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  • DOI: https://doi.org/10.1007/s11075-012-9573-y

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