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Identities and relations for Hermite-based Milne–Thomson polynomials associated with Fibonacci and Chebyshev polynomials

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The aim of this paper is to give many new and interesting identities, relations, and combinatorial sums including the Hermite-based Milne-Thomson type polynomials, the Chebyshev polynomials, the Fibonacci-type polynomials, trigonometric type polynomials, the Fibonacci numbers, and the Lucas numbers. By using Wolfram Mathematica version 12.0, we give surfaces graphics and parametric plots for these polynomials and generating functions. Moreover, by applying partial derivative operators to these generating functions, some derivative formulas for these polynomials are obtained. Finally, suitable connections of these identities, formulas, and relations of this paper with those in earlier and future studies are designated in detail remarks and observations.

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References

  1. Araci, S., Riyasat, M., Khan, S., Wani, S.A.: Some unified formulas involving generalized-Apostol-type-Gould–Hopper polynomials and multiple power sums. J. Math. Comput. Sci. 19(2), 97–115 (2019)

    Article  Google Scholar 

  2. Araci, S., Riyasat, M., Wani, S.A., Khan, S.: A new class of Hermite-Apostol type Frobenius–Euler polynomials and its applications. Symmetry 10(1), 1–16 (2018)

    MATH  Google Scholar 

  3. Benjamin, A.T., Ericksen, L., Jayawant, P., Shattuck, M.: Combinatorial trigonometry with Chebyshev polynomials. J. Stat. Plann. Inference 140(8), 2157–2160 (2010)

    Article  MathSciNet  Google Scholar 

  4. Boussayoud, A., Boughaba, S., Kerada, M., Araci, S., Acikgoz, M.: Generating functions of binary products of \(k\)-Fibonacci and orthogonal polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 2575–2586 (2019)

    Article  MathSciNet  Google Scholar 

  5. 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 

  6. Comtet, L.: Advanced Combinatorics. D. Reidel Publication Company, Dordrecht-Holland, Boston (1974)

  7. Costabile, F.A., Gualtieri, M.I., Napoli, A.: Polynomial sequences: elementary basic methods and application hints. A survey. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 3829–3862 (2019)

    Article  MathSciNet  Google Scholar 

  8. Dattoli, G., Lorenzutta, S., Maino, G., Torre, A., Cesarano, C.: Generalized Hermite polynomials and supergaussian forms. J. Math. Anal. Appl. 203, 597–609 (1996)

    Article  MathSciNet  Google Scholar 

  9. Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G., Torre, A.: Theory of generalized Hermite polynomials. Comput. Math. Appl. 28(4), 71–83 (1994)

    Article  MathSciNet  Google Scholar 

  10. Gould, H.W.: Table for Fundamentals of Series: Part I: Basic Properties of Series and Products, Vol.1 August 19 (2011). https://math.wvu.edu/~126hgould/Vol.1.PDF

  11. Fox, L., Parker, I.B.: Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London (1968)

    MATH  Google Scholar 

  12. Khan, S., Nahid, T., Riyasat, M.: Partial derivative formulas and identities involving \(2\)-variable Simsek polynomials. Bol. Soc. Mat. Mex. 26, 1–13 (2020)

    Article  MathSciNet  Google Scholar 

  13. Kilar, N., Simsek, Y.: Relations on Bernoulli and Euler polynomials related to trigonometric functions. Adv. Stud. Contemp. Math. 29(2), 191–198 (2019)

    MATH  Google Scholar 

  14. Kilar, N., Simsek, Y.: Two parametric kinds of Eulerian-type polynomials associated with Euler’s formula. Symmetry 11(9), 1097, 1–19 (2019)

  15. Kilar, N., Simsek, Y.: Some classes of generating functions for generalized Hermite- and Chebyshev-type polynomials: analysis of Euler’s formula. arXiv:1907.03640v1, 1–31 (2019)

  16. Kilar, N., Simsek, Y.: A note on Hermite-based Milne Thomson type polynomials involving Chebyshev polynomials and other polynomials. Techno-Science 3(1), 8–14 (2020)

    Google Scholar 

  17. Kilar, N., Simsek, Y.: Identities for special numbers and polynomials involving Fibonacci-type polynomials and Chebyshev polynomials. Adv. Stud. Contemp. Math. 30(4), 493–502 (2020)

    Google Scholar 

  18. Kilar, N., Simsek, Y.: A special approach to derive new formulas for some special numbers and polynomials. Turk. J. Math. 44, 2217–2240 (2020). https://doi.org/10.3906/mat-2005-116

    Article  MathSciNet  Google Scholar 

  19. Kim, T., Ryoo, C.S.: Some identities for Euler and Bernoulli polynomials and their zeros. Axioms 7(3), 56, 1–19 (2018)

  20. Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001)

    Book  Google Scholar 

  21. Koshy, T.: Pell and Pell-Lucas Numbers with Applications. Springer, New York (2014)

    Book  Google Scholar 

  22. Lebedev, N.N.: Special Functions and Their Applications. Revised English edition translated and edited by Richard A. Silverman, Prentice-Hall, Inc. Englewood Cliffs, N.J. (1965)

  23. Masjed-Jamei, M., Koepf, W.: Symbolic computation of some power-trigonometric series. J. Symb. Comput. 80, 273–284 (2017)

    Article  MathSciNet  Google Scholar 

  24. Masjed-Jamei, M., Beyki, M.R., Koepf, W.: A new type of Euler polynomials and numbers. Mediterr. J. Math. 15(138), 1–17 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Masjed-Jamei, M., Beyki, M.R., Omey, E.: On a parametric kind of Genocchi polynomials. J. Inequal. Spec. Funct. 9(2), 68–81 (2018)

    MathSciNet  Google Scholar 

  26. Masjed-Jamei, M., Beyki, M.R., Koepf, W.: An extension of the Euler–Maclaurin quadrature formula using a parametric type of Bernoulli polynomials. Bull. Sci. Math. 156(102798), 1–26 (2019)

    MathSciNet  MATH  Google Scholar 

  27. Moldovan, C.L., Păltănea, R.: Second degree Schoenberg operators with knots at the roots of Chebyshev polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 2793–2804 (2019)

  28. Ozdemir, G., Simsek, Y.: Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers. Filomat 30(4), 969–975 (2016)

    Article  MathSciNet  Google Scholar 

  29. Ozdemir, G., Simsek, Y., Milovanović, G.V.: Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials. Mediterr. J. Math. 14(117), 1–17 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Rassias, T.M., Srivastava, H.M., Yanushauskas, A.: Topics in Polynomials of One and Several Variables and Their Applications. World Scientific, Singapore (1991)

    Google Scholar 

  31. Riyasat, M., Khan, S.: Some results on \(q\)-Hermite based hybrid polynomials. Glas. Mat. 53(73), 9–31 (2018)

    Article  MathSciNet  Google Scholar 

  32. Riyasat, M., Khan, S.: A determinant approach to \(q\)-Bessel polynomials and applications. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 1571–1583 (2019)

    Article  MathSciNet  Google Scholar 

  33. Ryoo, C.S.: Differential equations arising from the \(3\)-variable Hermite polynomials and computation of their zeros. Differential Equations-Theory and Current Research Capter 5, editor: T.E. Moschandreou, (2018) https://doi.org/10.5772/intechopen.74355

  34. Ryoo, C.S., Khan, W.A.: On two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials. Mathematics 8(3), 417, 1–18 (2020)

  35. Simsek, Y.: Formulas for Poisson–Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and \( p \)-adic integral approach. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(2), 931–948 (2019)

  36. Simsek, Y., Cakic, N.: Identities associated with Milne-Thomson type polynomials and special numbers. J. Inequal. Appl. 2018:84, 1–13 (2018) https://doi.org/10.1186/s13660-018-1679-x

  37. Srivastava, H.M., Masjed-Jamei, M., Beyki, M.R.: A parametric type of the Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Appl. Math. Inf. Sci. 12(5), 907–916 (2018)

    Article  MathSciNet  Google Scholar 

  38. Srivastava, H.M., Masjed-Jamei, M., Beyki, M.R.: Some new generalizations and applications of the Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Rock. Mt. J. Math. 49(2), 681–697 (2019)

    Article  MathSciNet  Google Scholar 

  39. Zhang, W.: Some identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Quart. 42(2), 149–154 (2004)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The paper was supported by Scientific Research Project Administration of Akdeniz University with Project Number: FDK-2020-5276. Due to some suggested references and also suggestions, the authors would like to thank all referees for their valuable comments.

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  1. Department of Mathematics, Faculty of Science, University of Akdeniz, 07058, Antalya, Turkey

    Neslihan Kilar & Yilmaz Simsek

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  1. Neslihan Kilar
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  2. Yilmaz Simsek
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Correspondence to Neslihan Kilar.

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Kilar, N., Simsek, Y. Identities and relations for Hermite-based Milne–Thomson polynomials associated with Fibonacci and Chebyshev polynomials. RACSAM 115, 28 (2021). https://doi.org/10.1007/s13398-020-00968-3

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