Research Article
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Year 2023, , 203 - 211, 30.06.2023
https://doi.org/10.47000/tjmcs.1242781

Abstract

References

  • Abramowitz, M., Stegun, I.A., Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Courier Corporation, USA, 1972.
  • Apostol, T.M., On the Lerch zeta function, Pacic J. Math., 1(1951), 161–167.
  • Apostol, T.M., Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1976.
  • Dattoli, G., Germano, B., Licciardi, S., Martinelli, M. R., Umbral methods and harmonic numbers, Axioms, 7(2018), 1–9.
  • Kuş, S., Tuglu, N., Kızılates¸, C., A study of harmonic Fibonacci polynomials associated with Bernoulli-F and Euler–Fibonacci polynomials, Indian J. Pure Appl. Math., to appear.
  • Luo, Q.M., Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order, Adv. Stud. Contemp. Math. (Kyungshang), 10(2005), 63–70.
  • Luo, Q.M., Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10(2006), 917–925.
  • Luo, Q.M., The multplication formulas for the Apostol-Bernoulli and Apostol Euler polynomials of higher order, Integ. Trans. and Special Func., 20(2009), 377–391.
  • Luo, Q.M., Srivastava, H.M., Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308(2005), 290-302.
  • Luo, Q.M., Srivastava, H.M., Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl., 51(2006), 631-642.
  • Özvatan, M., Generalized Golden-Fibonacci Calculus and Applications, Master Thesis, The Graduate School of Engineering and Sciences of Izmir Institute of Technology, ˙Izmir, 2018.
  • Pashaev, O.K., Nalci, S., Golden quantum oscillator and Binet Fibonacci calculus, J Phys A: Math Theor., 45(2012), 015303.
  • Srivastava, H.M., Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc., 129(2000), 77-84.
  • Srivastava, H.M., Pinter, A., Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17(2004), 375-380.
  • Srivastava, H.M., Choi, J., Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001.
  • Tuglu N., Kızılates¸ C., Kesim, S., On the harmonic and hyperharmonic Fibonacci numbers, Advances in Difference Equations, 2015(2015), 297.

Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions

Year 2023, , 203 - 211, 30.06.2023
https://doi.org/10.47000/tjmcs.1242781

Abstract

In this article, the Apostol Bernoulli-Fibonacci polynomials are defined and various properties of Apostol Bernoulli-Fibonacci polynomials are obtained. Furthermore, Apostol Euler-Fibonacci numbers and polynomials are found. In addition, harmonic-based F exponential generating functions are defined for Apostol Bernoulli-Fibonacci numbers and Apostol Euler-Fibonacci numbers.

References

  • Abramowitz, M., Stegun, I.A., Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Courier Corporation, USA, 1972.
  • Apostol, T.M., On the Lerch zeta function, Pacic J. Math., 1(1951), 161–167.
  • Apostol, T.M., Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1976.
  • Dattoli, G., Germano, B., Licciardi, S., Martinelli, M. R., Umbral methods and harmonic numbers, Axioms, 7(2018), 1–9.
  • Kuş, S., Tuglu, N., Kızılates¸, C., A study of harmonic Fibonacci polynomials associated with Bernoulli-F and Euler–Fibonacci polynomials, Indian J. Pure Appl. Math., to appear.
  • Luo, Q.M., Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order, Adv. Stud. Contemp. Math. (Kyungshang), 10(2005), 63–70.
  • Luo, Q.M., Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10(2006), 917–925.
  • Luo, Q.M., The multplication formulas for the Apostol-Bernoulli and Apostol Euler polynomials of higher order, Integ. Trans. and Special Func., 20(2009), 377–391.
  • Luo, Q.M., Srivastava, H.M., Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308(2005), 290-302.
  • Luo, Q.M., Srivastava, H.M., Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl., 51(2006), 631-642.
  • Özvatan, M., Generalized Golden-Fibonacci Calculus and Applications, Master Thesis, The Graduate School of Engineering and Sciences of Izmir Institute of Technology, ˙Izmir, 2018.
  • Pashaev, O.K., Nalci, S., Golden quantum oscillator and Binet Fibonacci calculus, J Phys A: Math Theor., 45(2012), 015303.
  • Srivastava, H.M., Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc., 129(2000), 77-84.
  • Srivastava, H.M., Pinter, A., Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17(2004), 375-380.
  • Srivastava, H.M., Choi, J., Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001.
  • Tuglu N., Kızılates¸ C., Kesim, S., On the harmonic and hyperharmonic Fibonacci numbers, Advances in Difference Equations, 2015(2015), 297.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Elif Gülal This is me 0000-0003-1145-246X

Naim Tuglu 0000-0002-7277-0034

Publication Date June 30, 2023
Published in Issue Year 2023

Cite

APA Gülal, E., & Tuglu, N. (2023). Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions. Turkish Journal of Mathematics and Computer Science, 15(1), 203-211. https://doi.org/10.47000/tjmcs.1242781
AMA Gülal E, Tuglu N. Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions. TJMCS. June 2023;15(1):203-211. doi:10.47000/tjmcs.1242781
Chicago Gülal, Elif, and Naim Tuglu. “Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions”. Turkish Journal of Mathematics and Computer Science 15, no. 1 (June 2023): 203-11. https://doi.org/10.47000/tjmcs.1242781.
EndNote Gülal E, Tuglu N (June 1, 2023) Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions. Turkish Journal of Mathematics and Computer Science 15 1 203–211.
IEEE E. Gülal and N. Tuglu, “Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions”, TJMCS, vol. 15, no. 1, pp. 203–211, 2023, doi: 10.47000/tjmcs.1242781.
ISNAD Gülal, Elif - Tuglu, Naim. “Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions”. Turkish Journal of Mathematics and Computer Science 15/1 (June 2023), 203-211. https://doi.org/10.47000/tjmcs.1242781.
JAMA Gülal E, Tuglu N. Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions. TJMCS. 2023;15:203–211.
MLA Gülal, Elif and Naim Tuglu. “Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions”. Turkish Journal of Mathematics and Computer Science, vol. 15, no. 1, 2023, pp. 203-11, doi:10.47000/tjmcs.1242781.
Vancouver Gülal E, Tuglu N. Apostol Bernoulli-Fibonacci Polynomials, Apostol Euler-Fibonacci Polynomials and Their Generating Functions. TJMCS. 2023;15(1):203-11.