Gould-Hopper Based Degenerate Truncated Bernoulli Polynomials

Year 2022, Volume: 10 Issue: 3, 125 - 134, 09.09.2022

Uğur Duran

https://doi.org/10.36753/mathenot.893469

Abstract

In this study, we consider the truncated degenerate Bernoulli polynomials based on the Gould-Hopper polynomials and examine diverse properties and
formulas covering addition formulas, correlations and derivation property. Then, we derive some interesting implicit summation formulas and symmetric
identities. Moreover, we define Gould-Hopper based truncated degenerate Bernoulli polynomials of order $r$ and give some of their properties and
relations.

Keywords

Degenerate exponential function, truncated exponential function, Bernoulli polynomials, Gould-Hopper polynomials, exponential generating function

References

• [1] Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Mathematica. 15, 51-88 (1975).
• [2] Cheon,G.-S.: A note on the Bernoulli and Euler polynomials. Applied Mathematics Letters. 16, 365-368 (2003).
• [3] Dattoli, G., Ceserano, C., Sacchetti, D.: A note on truncated polynomials. Applied Mathematics and Computation. 134, 595-605 (2003).
• [4] Duran,U,Acikgoz,M.:GeneralizedGould-HopperbasedfullydegeneratecentralBellpolynomials.TurkishJournalof Analysis and Number Theory. 7 (5), 124-134 (2019).
• [5] Duran, U., Araci, S., Acikgoz, M.: Bell-Based Bernoulli polynomials with applications. Axioms. 10, no. 29 (2021).
• [6] Duran, U., Sadjang, P.N.: On Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q-parameter. Mathematics. 7, no. 121 (2019).
• [7] Duran, U., Acikgoz, M.: On generalized degenerate Gould-Hopper based fully degenerate Bell polynomials. Journal of Mathematics and Computer Science. 21 (3), 243-257 (2020).
• [8] Duran, U., Acikgoz, M.: On Degenerate Truncated Special Polynomials. Mathematics. 8 (1), no. 144 (2020).
• [9] Hassen, A., Nguyen, H.D.: Hypergeometric Bernoulli polynomials and Appell sequences. International Journal of Number Theory. 4, 767-774 (2008).
• [10] Howard, F.T.: Explicit formulas for degenerate Bernoulli numbers. Discrete Mathematics. 162, 175-185 (1996).
• [11] Kim, T., Kim, D.S., Jang, L.-C., Kwon, H.I.: Extended degenerate stirling numbers of the second kind and extended degenerate Bell polynomials. Utilitas Mathematica. 106, 11-21 (2018).
• [12] Kim, T., Yao, Y., Kim, D.S., Jang, G.-W.: Degenerate r-Stirling numbers and r-Bell polynomials. Russian Journal of Mathematical Physics. 25, 44-58 (2018).
• [13] Kim, T.: A note on degenerate Stirling polynomials of the second kind. Proceedings of the Jangjeon Mathematical Society. 20, 319-331 (2017).
• [14] Kurt, B: Explicit relations for the modified degenerate Apostol-type polynomials. Journal of Balikesir University Institute of Science and Technology. 20, 401-412 (2018).
• [15] Pathan, M.A., Khan, W.A.: Some implicit summation formulas and symmetric identities for the generalized Hermite- Bernoulli polynomials. Mediterranean Journal of Mathematics. 12, 679-695 (2015).
• [16] Srivastava, H.M., Araci, S., Khan, W.A., Acikgoz, M.: A note on the truncated-exponential based Apostol-type polynomials. Symmetry. 11, no. 538 (2019).
• [17] Srivastava, H.M., Choi, J.: Zeta and q-Zeta functions and associated series and integrals. Elsevier Science Publishers. Amsterdam (2012).
• [18] Srivastava, H.M., Pinter, A.: Remarks on some relationships between the Bernoulli and Euler polynomials. Applied Mathematics Letters. 17, 375-380 (2004).