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A NOTE ON HERMITE-BASED MILNE THOMSON TYPE POLYNOMIALS INVOLVING CHEBYSHEV POLYNOMIALS AND OTHER POLYNOMIALS

Year 2020, Volume: 3 Issue: 1, 8 - 14, 26.02.2020

Neslıhan Kılar Yılmaz Şimşek

Abstract

The aim of this paper is to investigate and survey some relations between new families of polynomials including r-parametric Hermite-based Milne Thomson type polynomials and other special numbers, the Bernoulli numbers, the Euler numbers, and the Chebyshev polynomials. By using generating functions and their functional equations of these polynomials are presented. Moreover, using Wolfram Mathematica 12.0 version, some plots and surface of these polynomials under the special conditions are shown. Finally, some remarks, comments and observations for these numbers and polynomials are given.

Keywords

Euler numbers and polynomials Bernoulli numbers and polynomials Chebyshev polynomials Hermite-based Milne Thomson type polynomials Generating function

Project Number

Project ID FDK 5276.

References

  • [1]. Abramowitz, M., Stegun, I. (1965). Handbook of mathematical functions with formulas, graphs, and mathematical tables, New York: Dover.
  • [2]. Carlitz, L. (1968). Bernoulli Numbers. Fibonacci Quarterly vol. 6, no. 3 p. 71–85.
  • [3]. Cesarano, C. (2012). Identities and genarating functions on Chebyshev polynomials. Georgian Mathematical Journal vol. 9, p. 427-440, DOI: 10.1515/gmj-2012-0031.
  • [4]. Cesarano, C. (2014). Generalized Chebyshev polynomials. Hacettepe Journal of Mathematics and Statistics, vol. 43, no. 5 p. 731-740.
  • [5]. Cesarano, C. (2015). Integral representations and new generating functions of Chebyshev polynomials. Hacettepe Journal of Mathematics and Statistics vol. 44, no. 3 p. 535-546.
  • [6]. Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G. & Torre, A. (1994). Theory of generalized Hermite polynomials. Computers & Mathematics with Applications vol. 28, no. 4, p. 71-83.
  • [7]. Dattoli, G., Lorenzutta, S., Maino, G., Torre, A. & Cesarano, C. (1996). Generalized Hermite polynomials and supergaussian forms. Journal of Mathematical Analysis and Applications vol. 203, p. 597-609.
  • [8]. Dattoli, G., Sacchetti, D., Cesarano, C. (2001). A note on Chebyshev polynomials. Annali dell’Universitá di Ferrara vol. 47, no. 1 p.107-115.
  • [9]. Fox, L., Parker, I.B. (1968). Chebyshev polynomials in numerical analysis. London: Oxford University Press.
  • [10]. Kilar, N., Simsek, Y. (2019a). Relations on Bernoulli and Euler polynomials related to trigonometric functions. Advanced Studies in Contemporary Mathematics vol. 29, no. 2 p. 191-198, http://dx.doi.org/10.17777/ascm2019.29.2.19.
  • [11]. Kilar, N., Simsek, Y. (2019b). Some classes of generating functions for generalized Hermite-and Chebyshev-type polynomials: Analysis of Euler’s formula. arXiv 2019, arxiv:1907.03640.
  • [12]. Kim, T., Ryoo, C. S. (2018). Some identities for Euler and Bernoulli polynomials and their zeros. Axioms, vol. 7, no.3, doi:10.3390/axioms7030056.
  • [13]. Lebedev, N.N. (1965). Special functions and their applications. Revised English edition translated and edited by Richard A. Silverman, Prentice-Hall, Inc. Englewood Cliffs, N.J. [14]. Masjed-Jamei, M., Koepf, W. (2017). Symbolic computation of some power trigonometric series. Journal of Symbolic Computation vol. 80, p. 273-284, http://dx.doi.org/10.1016/j.jsc.2016.03.004.
  • [15]. Masjed-Jamei, M., Beyki, M.R., Koepf, W. (2018). A new type of Euler polynomials and numbers. Mediterranean Journal of Mathematics vol. 15, no. 138 p. 1-17, https://doi.org/10.1007/s00009-018-1181-1.
  • [16]. Ozden, H., Simsek, Y., Srivastava, H.M. (2010). A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Computer Mathematics and its Applications vol. 60, p. 2779-2787, https://doi.org/10.1016/j.camwa.2010.09.031.
  • [17]. Ryoo, C.S. (2018). 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, http://dx.doi.org/10.5772/intechopen.74355.
  • [18]. Simsek, Y. (2019). Formulas for Poisson-Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach. RACSAM. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas vol. 113, no. 2, 931-948, https://doi.org/10.1007/s13398-018-0528-6.
  • [19]. Simsek, Y., Cakic, N. (2018). Identities associated with Milne-Thomson type polynomials and special numbers. Journal of Inequalities and Applications, vol. 2018, no. 84 p. 1-13, https://doi.org/10.1186/s13660-018-1679-x.
  • [20]. Srivastava, H.M. Masjed-Jamei, M., Beyki, M.R. (2018). A Parametric type of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. Applied Mathematics & Information Sciences vol. 12, no. 5 p. 907-916, http://dx.doi.org/10.18576/amis/120502.

Year 2020, Volume: 3 Issue: 1, 8 - 14, 26.02.2020

Neslıhan Kılar Yılmaz Şimşek

Abstract

Supporting Institution

The paper was supported by Scientific Research Project Administration of Akdeniz University Project ID FDK 5276.

Project Number

Project ID FDK 5276.

Thanks

Akdeniz University

References

  • [1]. Abramowitz, M., Stegun, I. (1965). Handbook of mathematical functions with formulas, graphs, and mathematical tables, New York: Dover.
  • [2]. Carlitz, L. (1968). Bernoulli Numbers. Fibonacci Quarterly vol. 6, no. 3 p. 71–85.
  • [3]. Cesarano, C. (2012). Identities and genarating functions on Chebyshev polynomials. Georgian Mathematical Journal vol. 9, p. 427-440, DOI: 10.1515/gmj-2012-0031.
  • [4]. Cesarano, C. (2014). Generalized Chebyshev polynomials. Hacettepe Journal of Mathematics and Statistics, vol. 43, no. 5 p. 731-740.
  • [5]. Cesarano, C. (2015). Integral representations and new generating functions of Chebyshev polynomials. Hacettepe Journal of Mathematics and Statistics vol. 44, no. 3 p. 535-546.
  • [6]. Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G. & Torre, A. (1994). Theory of generalized Hermite polynomials. Computers & Mathematics with Applications vol. 28, no. 4, p. 71-83.
  • [7]. Dattoli, G., Lorenzutta, S., Maino, G., Torre, A. & Cesarano, C. (1996). Generalized Hermite polynomials and supergaussian forms. Journal of Mathematical Analysis and Applications vol. 203, p. 597-609.
  • [8]. Dattoli, G., Sacchetti, D., Cesarano, C. (2001). A note on Chebyshev polynomials. Annali dell’Universitá di Ferrara vol. 47, no. 1 p.107-115.
  • [9]. Fox, L., Parker, I.B. (1968). Chebyshev polynomials in numerical analysis. London: Oxford University Press.
  • [10]. Kilar, N., Simsek, Y. (2019a). Relations on Bernoulli and Euler polynomials related to trigonometric functions. Advanced Studies in Contemporary Mathematics vol. 29, no. 2 p. 191-198, http://dx.doi.org/10.17777/ascm2019.29.2.19.
  • [11]. Kilar, N., Simsek, Y. (2019b). Some classes of generating functions for generalized Hermite-and Chebyshev-type polynomials: Analysis of Euler’s formula. arXiv 2019, arxiv:1907.03640.
  • [12]. Kim, T., Ryoo, C. S. (2018). Some identities for Euler and Bernoulli polynomials and their zeros. Axioms, vol. 7, no.3, doi:10.3390/axioms7030056.
  • [13]. Lebedev, N.N. (1965). Special functions and their applications. Revised English edition translated and edited by Richard A. Silverman, Prentice-Hall, Inc. Englewood Cliffs, N.J. [14]. Masjed-Jamei, M., Koepf, W. (2017). Symbolic computation of some power trigonometric series. Journal of Symbolic Computation vol. 80, p. 273-284, http://dx.doi.org/10.1016/j.jsc.2016.03.004.
  • [15]. Masjed-Jamei, M., Beyki, M.R., Koepf, W. (2018). A new type of Euler polynomials and numbers. Mediterranean Journal of Mathematics vol. 15, no. 138 p. 1-17, https://doi.org/10.1007/s00009-018-1181-1.
  • [16]. Ozden, H., Simsek, Y., Srivastava, H.M. (2010). A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Computer Mathematics and its Applications vol. 60, p. 2779-2787, https://doi.org/10.1016/j.camwa.2010.09.031.
  • [17]. Ryoo, C.S. (2018). 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, http://dx.doi.org/10.5772/intechopen.74355.
  • [18]. Simsek, Y. (2019). Formulas for Poisson-Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach. RACSAM. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas vol. 113, no. 2, 931-948, https://doi.org/10.1007/s13398-018-0528-6.
  • [19]. Simsek, Y., Cakic, N. (2018). Identities associated with Milne-Thomson type polynomials and special numbers. Journal of Inequalities and Applications, vol. 2018, no. 84 p. 1-13, https://doi.org/10.1186/s13660-018-1679-x.
  • [20]. Srivastava, H.M. Masjed-Jamei, M., Beyki, M.R. (2018). A Parametric type of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. Applied Mathematics & Information Sciences vol. 12, no. 5 p. 907-916, http://dx.doi.org/10.18576/amis/120502.
There are 19 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Original Research Articles
Authors

Neslıhan Kılar This is me 0000-0001-5797-6301

Yılmaz Şimşek 0000-0002-0611-7141

Project Number Project ID FDK 5276.
Publication Date February 26, 2020
Acceptance Date February 26, 2020
Published in Issue Year 2020 Volume: 3 Issue: 1

Cite

APA Kılar, N., & Şimşek, Y. (2020). A NOTE ON HERMITE-BASED MILNE THOMSON TYPE POLYNOMIALS INVOLVING CHEBYSHEV POLYNOMIALS AND OTHER POLYNOMIALS. Scientific Journal of Mehmet Akif Ersoy University, 3(1), 8-14.
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