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

Year 2023, Volume: 15 Issue: 1, 203 - 211, 30.06.2023

Elif Gülal Naim Tuglu

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.

Keywords

Bernoulli polynomials, Apostol polynomials, Apostol Bernoulli polynomials

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.