Some identities involving degenerate Stirling numbers arising from normal ordering

Taekyun Kim; Dae San Kim; Hye Kyung Kim; Taekyun Kim; Dae San Kim; Hye Kyung Kim

doi:10.3934/math.2022956

AIMS Mathematics

2022, Volume 7, Issue 9: 17357-17368. doi: 10.3934/math.2022956

Previous Article Next Article

Research article

Some identities involving degenerate Stirling numbers arising from normal ordering

Taekyun Kim ^{1
,
,},
Dae San Kim ^{2
,
,},
Hye Kyung Kim ^{3
,
,}

1.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea
2.
Department of Mathematics, Sogang University, Seoul 121-742, Korea
3.
Department of Mathematics Edunction, Daegu Catholic University, Gyeonsang 38430, Korea

Received: 25 April 2022 Revised: 04 July 2022 Accepted: 14 July 2022 Published: 26 July 2022
MSC : 11B73, 11B83

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind, which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal orderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind.
Citation: Taekyun Kim, Dae San Kim, Hye Kyung Kim. Some identities involving degenerate Stirling numbers arising from normal ordering[J]. AIMS Mathematics, 2022, 7(9): 17357-17368. doi: 10.3934/math.2022956

Related Papers:

Abstract

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind, which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal orderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind.

References

[1]	S. Araci, A new class of Bernoulli polynomials attached to polyexponential functions and related identities, Adv. Stud. Contemp. Math. (Kyungshang), 31 (2021), 195–204.
[2]	P. Blasiak, Combinatorics of Boson normal ordering and some applications, Ph.D. Thesis, University of Paris, Paris, 2005.
[3]	P. Blasiak, K. A. Penson, A. I. Solomon, The general Boson normal ordering problem, Phys. Lett. A, 309 (2003), 198–205. https://doi.org/10.1016/S0375-9601(03)00194-4 doi: 10.1016/S0375-9601(03)00194-4
[4]	L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15 (1979), 51–88.
[5]	L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Co., Dordrecht, 1974.
[6]	B. S. El-Desouky, Multiparameter non-central Stirling numbers, Fibonacci Quart., 32 (1994), 218–225.
[7]	B. S. El-Desouky, N. P. Cakic, T. Mansour, Modified approach to generalized Stirling numbers via differential operators, Appl. Math. Lett., 23 (2010), 115–120. https://doi.org/10.1016/j.aml.2009.08.018 doi: 10.1016/j.aml.2009.08.018
[8]	B. S. El-Desouky, N. A. El-Bedwehy, A. Mustafa, F. M. A. Menem, A family of generalized Stirling numbers of the first kind, Appl. Math., 5 (2014), 1573–1585. https://doi.org/10.4236/am.2014.510150 doi: 10.4236/am.2014.510150
[9]	T. Kim, D. S. Kim, Degenerate $r$-Whitney numbers and degenerate $r$-Dowling polynomials in via boson operators, Adv. Appl. Math., 140 (2022). https://doi.org/10.1016/j.aam.2022.102394
[10]	T. Kim, D. S. Kim, H. K. Kim, Normal ordering of degenerate integral power of number operator and its applications, Appl. Math. Sci. Eng., 30 (2022), 440–447. https://doi.org/10.1080/27690911.2022.2083120 doi: 10.1080/27690911.2022.2083120
[11]	D. S. Kim, T. Kim, A note on a new type of degenerate Bernoulli numbers, Russ. J. Math. Phys., 27 (2020), 227–235. https://doi.org/10.1134/S1061920820020090 doi: 10.1134/S1061920820020090
[12]	T. Kim, D. S. Kim, On some degenerate differential and degenerate difference operator, Russ. J. Math. Phys., 29 (2022), 37–46. https://doi.org/10.1134/S1061920822010046 doi: 10.1134/S1061920822010046
[13]	A. Perelomov, Generalized coherent states and their applications, Texts and Monographs in Physics, SpringerVerlag, Berlin, 1986.
[14]	S. Roman, The umbral calculus, Pure and Applied Mathematics, Academic Press, New York, 1984.
[15]	Y. Simsek, Construction of generalized Leibnitz type numbers and their properties, Adv. Stud. Contemp. Math. (Kyungshang), 31 (2021), 311–323. http://doi.org/10.17777/ascm2021.31.3.311 doi: 10.17777/ascm2021.31.3.311

Reader Comments

Your name:*

Email:*
© 2022 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)