Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials

Abstract

The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $\lambda$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $\lambda$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the $\lambda$-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the $\lambda$-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour some randomly selected special cases of their parameters. Finally, we conclude the paper with some comments and observations on our results.

Keywords

• Generating functions
• Functional Equations
• Special numbers and polynomials
• Stirling numbers of the first kind
• Apostol-type numbers and polynomials
• Simsek numbers and polynomials
• Bernoulli numbers of the second kind
• Daehee numbers and polynomials
• Integral formulas
• Chu-Vandermonde identity
• Bernstein basis functions
• Combinatorial sums

References

1. Acikgoz, M., Araci, S., On generating function of the Bernstein polynomials, AIP Conf. Proc., 1281(1) (2010), 1141. https://doi.org/10.1063/1.3497855
2. Bona, M., Introduction to Enumerative Combinatorics, The McGraw-Hill Companies Inc., New York, 2007.
3. Charalambides, C. A., Enumerative Combinatorics, Chapman and Hall/ CRC Press Company, London, 2002.
4. Choi, J., Note on Apostol-Daehee polynomials and numbers, Far East J. Math. Sci., 101(8) (2017), 1845-1857. https://doi.org/10.17654/MS101081845
5. Comtet, L., Advanced Combinatorics, D. Reidel Publication Company, Dordrecht-Holland/Boston-U.S.A., 1974.
6. Jordan, C., Calculus of Finite Differences (2nd ed.), Chelsea Publishing Company, New York, 1950.
7. El-Desouky, B. S., Mustafa, A., New results and matrix representation for Daehee and Bernoulli numbers and polynomials, Appl. Math. Sci. (Ruse) 9(73) (2015), 3593-3610. https://doi.org/10.12988/ams.2015.53282
8. Cakic, N. P., Milovanovic, G. V., On generalized Stirling numbers and polynomials, Mathematica Balkanica 18 (2004), 241-248.

9. Duman, O., Nonlinear Bernstein-type operators providing a better error estimation, Miskolc Math. Notes, 15(2) (2014), 393-400. https://doi.org/10.18514/MMN.2014.1136
10. Erkus, E., Duman, O., Srivastava, H. M., Statistical approximation of certain positive linear operators constructed by means of the Chan-Chyan-Srivastava polynomials, Appl. Math. Comput., 182 (2006), 213-222. https://doi.org/10.1016/j.amc.2006.01.090
11. Kim, D. S., Kim, T., Daehee numbers and polynomials, Appl. Math. Sci. (Ruse), 7 (2013), 5969-5976. https://doi.org/10.12988/ams.2013.39535
12. Kim, D. S., Kim, T., Lee, S.-H., Seo, J.-J., A Note on the lambda-Daehee Polynomials, Int. J. Math. Anal, 7(62) (2013), 3069-3080. https://doi.org/10.12988/ijma.2013.311264
13. Kim, T., Kim, D. S., Dolgy, D. V., Seo J.-J., Bernoulli polynomials of the second kind and their identities arising from umbral calculus, J. Nonlinear Sci. Appl. 9 (2016), 860-869. https://doi.org/10.22436/jnsa.009.03.14
14. Kucukoglu, I., Simsek, Y., On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers, Appl. Anal. Discrete Math., 13 (2019), 478-494. https://doi.org/10.2298/AADM180215016K
15. Kucukoglu, I., Simsek, B., Simsek, Y., Generating functions for new families of combinatorial numbers and polynomials: approach to Poisson-Charlier polynomials and probability distribution function, Axioms, 8(4) (2019), 112. https://doi.org/10.3390/axioms8040112
16. Lorentz, G. G., Bernstein Polynomials, Chelsea Pub. Comp., New York, NY, USA, 1986.
17. Merlini, D., Sprugnoli, R., Verri, M. C., The Cauchy numbers, Discrete Math., 306(16) (2006), 1906-1920. https://doi.org/10.1016/j.disc.2006.03.065
18. Qi, F., Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat, 28(2) (2014), 319-327. https://doi.org/10.2298/ FIL1402319O
19. Park, J.-W., On the λ-Daehee polynomials with q-parameter, J. Comput. Anal. Appl., 20(1) (2016), 11-20.
20. Roman, S., The Umbral Calculus, Dover Publ. Inc., New York, 2005.
21. Simsek, Y., Functional equations from generating functions: A novel approach to deriving identities for the Bernstein basis functions, Fixed Point Theory Appl., 80 (2013), 1-13. https://doi.org/10.1186/1687-1812-2013-80
22. Simsek, Y., Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl., 87 (2013), 343-355. https://doi.org/10.1186/1687-1812-2013-87
23. Simsek, Y., Unification of the Bernstein-type polynomials and their applications, Bound. Value Probl., 56 (2013), 1-15. https://doi.org/10.1186/1687-2770-2013-56
24. Simsek, Y., Generating functions for the Bernstein type polynomials: A new approach to deriving identities and applications for the polynomials, Hacet. J. Math. Stat., 43(1) (2014), 1-14.
25. Simsek, Y., Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers, Math. Methods Appl. Sci., 38(14) (2015), 3007-3021. https://doi.org/10.1002/mma.3276
26. Simsek, Y., Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications, Cogent Math. Stat. 1269393 (2016). https://doi.org/10.1080/23311835.2016.1269393
27. Simsek, Y., Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 26(3) (2016), 555-566.
28. Simsek, Y., Identities on the Changhee numbers and Apostol-type Daehee polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 27(2) (2017), 199-212. https://doi.org/10.23001/ascm2017.27.2.199
29. Simsek, Y., Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals, Turk. J. Math., 42 (2018), 557-577. https: //doi.org/10.3906/mat-1703-114
30. Simsek, Y., Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus J. Pure Appl. Math., 1(1) (2019), 1-76, Article ID: MTJPAM-D-19-00005.
31. Simsek, Y., Acikgoz, M., A new generating function of (q-) Bernstein-type polynomials and their interpolation function, Abstr. Appl. Anal., 769095 (2010). https://doi.org/10.1155/ 2010/769095
32. Simsek, Y., Yardimci, A., Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals, Adv. Difference Equ., 308 (2016). https://doi.org/10.1186/s13662-016-1041-x
33. Simsek, Y., Kucukoglu, I., Some Certain Classes of Combinatorial Numbers and Polynomials Attached to Dirichlet Characters: Their Construction by p-adic Integration and Applications to Probability Distribution Functions, A Chapter in the Book: Mathematical Analysis in Interdisciplinary Research, Th. M. Rassias, I. N. Parasidis and E. Providas (Eds.), Springer International Publishing, Springer Nature Switzerland AG (2021), In press.
34. Srivastava, H. M., Kucukoglu, I., Simsek, Y., Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials, J. Number Theory, 181 (2017), 117-146. https://doi.org/10.1016/j.jnt.2017.05.008
35. Wolfram Research Inc., Mathematica Online (Wolfram Cloud), Champaign, IL, 2020. https://www.wolframcloud.com