A note on degenerate derangement polynomials and numbers

Taekyun Kim; Dae San Kim; Hyunseok Lee; Lee-Chae Jang; Taekyun Kim; Dae San Kim; Hyunseok Lee; Lee-Chae Jang

doi:10.3934/math.2021380

AIMS Mathematics

2021, Volume 6, Issue 6: 6469-6481. doi: 10.3934/math.2021380

Previous Article Next Article

Research article Special Issues

A note on degenerate derangement polynomials and numbers

1.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2.
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
3.
Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea

Received: 17 November 2021 Accepted: 12 April 2021 Published: 15 April 2021
MSC : 11B73, 11B83, 60G50

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.
Citation: Taekyun Kim, Dae San Kim, Hyunseok Lee, Lee-Chae Jang. A note on degenerate derangement polynomials and numbers[J]. AIMS Mathematics, 2021, 6(6): 6469-6481. doi: 10.3934/math.2021380

Related Papers:

Abstract

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.

References

[1]	L. Carlitz, The number of derangements of a sequence with given specification, Fibonacci Quart., 16 (1978), 255–258.
[2]	D. V. Dolgy, D. S. Kim, T. Kim, J. Kwon, On fully degenerate Bell numbers and polynomials, Filomat, 2020. Available from: http://journal.pmf.ni.ac.rs/filomat/index.php/filomat/article/view/10239.
[3]	D. M. Jackson, Laguerre polynomials and derangements, Math. Proc. Cambridge Philos. Soc., 80 (1976), 213–214. doi: 10.1017/S030500410005283X
[4]	A. W. Joseph, A problem in derangements, J. Inst. Actuaries Students' Soc., 6 (1946), 14–22.
[5]	W. A. Khan, A new class of degenerate Frobenius-Euler-Hermite polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 28 (2018), 567–576.
[6]	N. Kilar, Y. Simsek, A new family of Fubini type numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials, J. Korean Math. Soc., 54 (2017), 1605–1621.
[7]	D. S. Kim, T. Kim, A note on a new type of degenerate Bernoulli numbers, Russ. J. Math. Phys., 27 (2020), 227–235. doi: 10.1134/S1061920820020090
[8]	T. Kim, D. S. Kim, Some identities on derangement and degenerate derangement polynomials, In: Advances in mathematical inequalities and applications, Singapore: Birkhäuser/Springer, 2018,265–277.
[9]	T. Kim, D. S. Kim, Note on the degenerate Gamma function, Russ. J. Math. Phys., 27 (2020), 352–358. doi: 10.1134/S1061920820030061
[10]	T. Kim, D. S. Kim, D. V. Dolgy, J. Kwon, Some identities of derangement numbers, Proc. Jangjeon Math. Soc., 21 (2018), 125–141.
[11]	T. Kim, D. S. Kim, G. W. Jang, A note on degenerate Fubini polynomials, Proc. Jangjeon Math. Soc., 20 (2017), 52–531.
[12]	T. Kim, D. S. Kim, G. W. Jang, J. Kwon, A note on some identities of derangement polynomials, J. Inequal. Appl., 2018 (2018), 40. doi: 10.1186/s13660-018-1636-8
[13]	T. Kim, D. S. Kim, H. I. Kwon, L. C. Jang, Fourier series of sums of products of $r$-derangement functions, J. Nonlinear Sci. Appl., 11 (2018), 575–590.
[14]	T. Kim, D. S. Kim, J. Kwon, H. Lee, A note on degenerate gamma random variables, Revista de edu., 388 (2020), 39–44.
[15]	D. S. Lee, H. K. Kim, On the new type of degenerate poly-Genocchi numbers and polynomials, Adv. Differ. Equ., 2020 (2020), 431. doi: 10.1186/s13662-020-02886-5
[16]	Y. Ma, T. Kim, A note on negative $\lambda$-binomial distribution, Adv. Differ. Equ., 2020 (2020), 569. doi: 10.1186/s13662-020-03030-z
[17]	S. Roman, The umbral calculus, New York: Academic Press, Inc., 1984.
[18]	S. M. Ross, Introduction to probability models, 11 Eds., Amsterdam: Elsevier/Academic Press, 2014.
[19]	S. K. Sharma, W. A. Khan, S. Araci, S. S. Ahmed, New construction of type 2 degenerate central Fubini polynomials with their certain properties, Adv. Differ. Equ., 2020 (2020), 587. doi: 10.1186/s13662-020-03055-4
[20]	Y. Simsek, Identities on the Changhee numbers and Apostol-type Daehee polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 27 (2017), 199–212.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)