Combinatorial Identities with Several Kinds of Degenerate-Daehee Sequences
In this paper, we mainly make use of the probabilistic method to calculate several different moment representations of the degenerate Daehee numbers of the third kind with degenerate log function. We also obtain the moment expressions of the degenerate Daehee numbers of higher-order and degenerate Daehee numbers of the second kind. When deriving the moment representations of degenerate Stirling numbers and the degenerate Bernoulli numbers, we arrive at the combinatorial identities of relationships of them and we prove them by the probabilistic method.
2. El-Desouky. B.S., Mustafa. A.: New Results on Higher-order Daehee and Bernoulli Numbers and Polynomials. Advances in Difference Equations,(32), 2016.
3. Jang G.W., Kwon J., Lee. J.G.: Some Identities of Degenerate Daehee Numbers Arising from Nonlinear Differential Equation. Advances in Difference Equations,(206), 2017.
4. Kim D.S., Kim T., Kwon H.I., Dolgy. D.V.: A Note on Degenerate Bernoulli Numbers and Polynomials Associated with P-adic Invariant Integral on Zp. Applied Mathematics and Computation, 295:198C204, 2015.
5. Kim D.S., Kim T., Kwon H.I.,, Jang G.W.: Degenerate Daehee Polynomials of the Second Kind. Proceedings of the Jangjeon Mathematical Society, 21(1):83C97, 2018.
6. Kim T.: On the Degenerate Cauchy Numbers and Polynomials. Proceedings of the Jangjeon Mathematical Society, 18(3):307C312, 2015.
7. Kim T., Kim D.S., Kwon H.I.: A Note on Degenerate Stirling Numbers and Their Applications. Proceedings of the Jangjeon Mathematical Society, 21(2):195C203, 2018.
8. Liu C., Wuyungaowa: Application of Probabilistic Method on Daehee Sequences. European Journal of Pure and Applied Mathematics, 11(1):69C78, 2018.
9. Pro S.S., Kim T., RIM S.H.: Identities of the Degenerate Daehee Numbers with the Bernoulli Numbers of the Second Kind Arising from Nonlinear Differential Equation. Journal of Nonlinear Sciences and Applications, 10(12):6219C6228, 2017.
10. Pro S.S., Kim T., RIM S.H.: Degenerate Daehee Numbers of the Third Kind. Bulletin of the Malaysian Mathematical Society, 6(11), 2018.
11. Sun P.: Moment Representation of Bernoulli Polynomial, Euler Polynomial and Gegenbauer Polynomials. Statistics and Probability Letters, 77:748C751, 2007.
12. Sun P., Wang T.: Probabilistic Representation with Application of Stirling Numbers. Acta Mathematica Sinica Chinese Series, 41(2):281C290, 1998.
13. Wilf. H.S.: Generating Functionology. Academic Press, New York, 1990.
Copyright (c) 2020 Journal of Progressive Research in Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
TRANSFER OF COPYRIGHT
JPRM is pleased to undertake the publication of your contribution to Journal of Progressive Research in Mathematics.
The copyright to this article is transferred to JPRM(including without limitation, the right to publish the work in whole or in part in any and all forms of media, now or hereafter known) effective if and when the article is accepted for publication thus granting JPRM all rights for the work so that both parties may be protected from the consequences of unauthorized use.