Journal of Applied Mathematics and Computation

DOI：http://dx.doi.org/10.26855/jamc.2023.06.002

Date: May 26,2023 Hits: 83

Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. In 2019, T. Kim and D. Kim defined the type 2 Daehee polynomials by the generating function of the type 2 Bernoulli polynomials and express the central factorial numbers of the second kind in terms of type 2 Bernoulli and type 2 Daehee numbers of negative integral orders. In this paper, we define the generating function of the high-order degenerate type 2 Daehee polynomials, then we study the high-order degenerate type 2 Daehee numbers and polynomials by using the method of generating function and Riordon array. First, applying generating functions methods, we obtain some character involving the high-order degenerate type 2 Daehee polynomials. In addition, we establish some new equations and relations involving two classes of generalized Stirling numbers, generalized Lah numbers, high-order type 2 Bernoulli polynomials, the central factorial numbers of the second kind, generalized Harmonic numbers and so on.

[1] Kim, T., Kim, D. S., Kim, H. Y., and Kwon, J. (2020). A new type degenerate Daehee numbers and polynomials. arXiv preprint arXiv:2004.08743.

[2] Pyo, S. S., Kim, T., and Rim, S. H. (2018). Degenerate Daehee numbers of the third kind. Mathematics, 6(11), 239.

[3] Kim, T., Kim, D. S., Kim, H. Y., and Kwon, J. (2020). Some results on degenerate Daehee and Bernoulli numbers and polynomials. Advances in Difference Equations, 2020, 1-13.

[4] Kumar Sharma, S., Khan, W. A., Araci, S., and Ahmed, S. S. (2020). New type of degenerate Daehee polynomials of the second kind. Advances in Difference Equations, 2020(1), 1-14.

[5] Kim, T., Kim, D. S. (2019). A note on type 2 Changhee and Daehee polynomials. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113, 2763-2771.

[6] Kim, D. S., Kim, T. (2015). Identities arising from higher-order Daehee polynomial bases. Open Mathematics, 13(1), 196-208.

[7] Chen, S. and Wuyungaowa (2019). A note on Degenerate Type 2 Changhee Polynomials and numbers. Journal of Progressive Research in Mathematics, 15, 2768-2780.

[8] Ma, M., and Lim, D. (2019). A Note on Type 2 ω-Daehee Polynomials. Mathematics, 7(8), 697.

[9] Kim, D. S., Kim, H. Y., Kim, D., and Kim, T. (2019). Identities of symmetry for type 2 Bernoulli and Euler polynomials. Sym-metry, 11(5), 613.

[10] Wang, W. (2010). Riordan arrays and harmonic number identities. Computers and Mathematics with Applications, 60(5), 1494-1509.

[11] Luo, Y. N., and Wuyungaowa, B. (2019). Some combinatorial identities about Daehee sequences. J. Comb. Math. Comb. Comput., 108, 75-87.

[12] Wuyungaowa, S. W. (2015). Sums of involving the Harmonic numbers and the Binomial coefficients. Amer. J. Comput. Math, 5, 96-105.

[13] Cheon, G. S., and El-Mikkawy, M. E. A. (2008). Generalized harmonic numbers with Riordan arrays. Journal of Number Theory, 128(2), 413-425.

[14] Kim, T. (2017). A note on degenerate Stirling polynomials of the second kind. arXiv preprint arXiv:1704.02290.

[15] Charalambides, C. A. (2005). Combinatorial methods in discrete distributions. John Wiley and Sons.

[16] Wuyungaowa, N. (2018). Some Identities Involving the Higher-Order Changhee Numbers and Polynomials. Journal of Applied Mathematics and Physics, 6(04), 647.

**Some Identities Involving the High-Order Degenerate Type 2 Daehee Polynomials**

**How to cite this paper:** Pengfei Zhang, Wuyungaowa. (2023) Some Identities Involving the High-Order Degenerate Type 2 Daehee Polynomials. *Journal of Applied Mathematics and Computation*, **7**(**2**), 211-223.

**DOI: http://dx.doi.org/10.26855/jamc.2023.06.002**

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