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A sequence of Appell polynomials and the associated Jakimovski–Leviatan operators

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Abstract

We are concerned with a special sequence of Appell polynomials, related to the Rényi and Tsallis entropies for the binomial distribution. The generating function is investigated: it is logarithmically convex and has remarkable connections with the modified Bessel function \(I_0(t)\) and with the index of coincidence for Poisson distribution. The specific form of the Appell polynomials leads to specific properties of the associated Jakimovski–Leviatan operators.

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References

  1. Abel, U., Ivan, M.: Asymptotic expansion of the Jakimovski–Leviatan operators and their derivatives. In: Functions, Series, Operators(Budapest, 1999), János Bolyai Math. Soc., Budapest, pp. 103–119 (2002)

  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, New York (1972)

    MATH  Google Scholar 

  3. Altomare, F., Campiti, M.: Korovkin-type Approximation Theory and Its Applications, de Gruyter Studies in Mathematics 17. Walter de Gruyter, Berlin (1994)

    Book  Google Scholar 

  4. Acu, A.M., Bascanbaz-Tunca, G., Raşa, I.: Bounds for indices of coincidence and entropies. Math. Inequl. Appl. (in press)

  5. Bretti, G., Cesarano, C., Ricci, P.E.: Laguerre-type exponentials and generalized Appell polynomials. Comput. Math. Appl. 48, 833–839 (2004)

    Article  MathSciNet  Google Scholar 

  6. Bretti, G., Ricci, P.E.: Multidimensional extensions of the Bernoulli and Appell polynomials. Taiwan J. Math. 8(3), 415–428 (2004)

    Article  MathSciNet  Google Scholar 

  7. Costabile, F.A., Longo, E.: A determinantal approach to Appell polynomials. J. Comput. Appl. Math. 234(5), 1528–1542 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Boston (1994)

    MATH  Google Scholar 

  10. Gupta, V., Rassias, M.T.: Moments of Linear Positive Operators and Approximation. Springer, New York (2019)

    Book  Google Scholar 

  11. Gupta, P., Acu, A.M., Agrawal, P.N.: Jakimovski-Leviatan operators of Kantorovich type involving multiple Appell polynomials. Georgian Math. J. 28(1), 73–82 (2021)

    Article  MathSciNet  Google Scholar 

  12. He, M.X., Ricci, P.E.: Differential equation of Appell polynomials via the factorization method. J. Comput. Appl. Math. 139, 231–237 (2002)

    Article  MathSciNet  Google Scholar 

  13. Ismail, M.E.H.: Remarks on differential equation of Appell polynomials via the factorization method. J. Comput. Appl. Math. 154, 243–245 (2003)

    Article  MathSciNet  Google Scholar 

  14. Jakimovski, A., Leviatan, D.: Generalized Szász operators for the approximation in the infinite interval. Mathematica (Cluj) 34, 97–103 (1969)

    MATH  Google Scholar 

  15. Mursaleen, M., Al-Abied, A., Acu, A.M.: Approximation by Chlodowsky type of Szasz operators based on Boas-Buck-type polynomials. Turk. J. Math. 42, 2243–2259 (2018)

    Article  MathSciNet  Google Scholar 

  16. Raşa, I.: Entropies and Heun functions associated with positive linear operators. Appl. Math. Comput. 268, 422–431 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Raşa, I.: Convexity properties of some entropies. Results Math. 73, 105 (2018)

    Article  MathSciNet  Google Scholar 

  18. Raşa, I.: Convexity properties of some entropies (II). Results Math. 74, 154 (2019)

    Article  MathSciNet  Google Scholar 

  19. Sucu, S., Içöz, G., Varma, S.: On some extensions of Szász operators including Boas-Buck-type polynomials. Abstract Appl Anal; Article ID 680340, p. 15 (2012)

  20. Varma, S., Sucu, S., Içöz, G.: Generalization of Szász operators involving Brenke type polynomials. Comput. Math. Appl. 64, 121–127 (2012)

    Article  MathSciNet  Google Scholar 

  21. https://math.stackexchange.com/questions/3210839/the-exponential-generating-function-for-the-central-binomial-coefficients

  22. https://oeis.org/

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Acknowledgements

We are very grateful to the reviewers for their highly valuable comments and suggestions which improved considerably the first version of the paper.

This work was supported by a Hasso Plattner Excellence Research Grant (LBUS-HPI-ERG-2020-04), financed by the Knowledge Transfer Center of the Lucian Blaga University of Sibiu.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No.5-7, 550012, Sibiu, Romania

    Ana-Maria Acu

  2. Department of Mathematics, Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, 400114, Cluj-Napoca, Romania

    Ioan Cristian Buscu & Ioan Rasa

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  1. Ana-Maria Acu
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  2. Ioan Cristian Buscu
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  3. Ioan Rasa
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Correspondence to Ana-Maria Acu.

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Acu, AM., Buscu, I.C. & Rasa, I. A sequence of Appell polynomials and the associated Jakimovski–Leviatan operators. Anal.Math.Phys. 11, 88 (2021). https://doi.org/10.1007/s13324-021-00525-0

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  • DOI: https://doi.org/10.1007/s13324-021-00525-0

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