A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

Waseem A. Khan; Abdulghani Muhyi; Rifaqat Ali; Khaled Ahmad Hassan Alzobydi; Manoj Singh; Praveen Agarwal; Waseem A. Khan; Abdulghani Muhyi; Rifaqat Ali; Khaled Ahmad Hassan Alzobydi; Manoj Singh; Praveen Agarwal

doi:10.3934/math.2021731

AIMS Mathematics

2021, Volume 6, Issue 11: 12680-12697. doi: 10.3934/math.2021731

Previous Article Next Article

Research article Special Issues

A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

1.
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia
2.
Department of Mathematics, Hajjah University, Hajjah, Yemen
3.
Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, P.O Box 9004, Postal Code:61413. Abha, Saudi Arabia
4.
Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia
5.
Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India
6.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman AE 346, United Arab Emirates
7.
International Center for Basic and Applied Sciences, Jaipur 302029, India

Received: 19 July 2021 Accepted: 18 August 2021 Published: 06 September 2021
MSC : 05A19, 11B73, 11B83

The main object of this article is to present type 2 degenerate poly-Bernoulli polynomials of the second kind and numbers by arising from modified degenerate polyexponential function and investigate some properties of them. Thereafter, we treat the type 2 degenerate unipoly-Bernoulli polynomials of the second kind via modified degenerate polyexponential function and derive several properties of these polynomials. Furthermore, some new identities and explicit expressions for degenerate unipoly polynomials related to special numbers and polynomials are obtained. In addition, certain related beautiful zeros and graphical representations are displayed with the help of Mathematica.
Citation: Waseem A. Khan, Abdulghani Muhyi, Rifaqat Ali, Khaled Ahmad Hassan Alzobydi, Manoj Singh, Praveen Agarwal. A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties[J]. AIMS Mathematics, 2021, 6(11): 12680-12697. doi: 10.3934/math.2021731

Related Papers:

Abstract

The main object of this article is to present type 2 degenerate poly-Bernoulli polynomials of the second kind and numbers by arising from modified degenerate polyexponential function and investigate some properties of them. Thereafter, we treat the type 2 degenerate unipoly-Bernoulli polynomials of the second kind via modified degenerate polyexponential function and derive several properties of these polynomials. Furthermore, some new identities and explicit expressions for degenerate unipoly polynomials related to special numbers and polynomials are obtained. In addition, certain related beautiful zeros and graphical representations are displayed with the help of Mathematica.

References

[1]	L. Carlitz, Degenerate stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15 (1979), 51–88.
[2]	L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math., 7 (1956), 28–33. doi: 10.1007/BF01900520
[3]	D. V. Dolgy, W. A. Khan, A note on type two degenerate poly-Changhee polynomials of the second kind, Symmetry, 13 (2021), 579. doi: 10.3390/sym13040579
[4]	W.A. Khan, A new class of degenerate Frobenius-Euler-Hermite polynomials, Adv. Stud. Contemp. Math., 28 (2018), 567–576.
[5]	W. A. Khan, M. Acikgoz, U. Duran, Note on the type 2 degenerate multi-poly-Euler polynomials, Symmetry, 12 (2020), 1691. doi: 10.3390/sym12101691
[6]	W. A. Khan, R. Ali, K. A. H. Alzobydi, N. Ahmed, A new family of degenerate poly-Genocchi polynomials with its certain properties, J. Funct. Spaces, 2021 (2021), 6660517.
[7]	D. S. Kim, T. Kim, A note on polyexponential and unipoly functions, Russ. J. Math. Phys., 26 (2019), 40–49. doi: 10.1134/S1061920819010047
[8]	T. Kim, A note on degenerate Stirling polynomials of the second kind, Proc. Jangjeon Math. Soc., 20 (2017), 319–331.
[9]	T. Kim, L. C. Jang, D. S. Kim, H. Y. Kim, Some identities on type 2 degenerate Bernoulli polynomials of the second kind, Symmetry, 12 (2020), 510. doi: 10.3390/sym12040510
[10]	T. Kim, D. S. Kim, Degenerate polyexponential functions and degenerate Bell polynomials, J. Math. Anal. Appl., 487 (2020), 124017. doi: 10.1016/j.jmaa.2020.124017
[11]	D. S. Kim, T. Kim, A note on a new type of degenerate Bernoulli numbers, Russ. J. Math. Phys. 27 (2020), 227–235.
[12]	T. Kim, D. S. Kim, Degenerate Laplace transform and degenerate gamma function, Russ. J. Math. Phys., 24 (2017), 241–248. doi: 10.1134/S1061920817020091
[13]	T. Kim, H. I. Kwon, S. H. Lee, J. J. Seo, A note on poly-Bernoulli polynomials of the second kind, Adv. Differ. Equ., 2014 (2014), 119. doi: 10.1186/1687-1847-2014-119
[14]	T. Kim, D. S. Kim, J. Kwon, H. Y. Kim, A note on degenerate Genocchi and poly-Genocchi numbers and polynomials, J. Inequal. Appl., 2020 (2020), 110. doi: 10.1186/s13660-020-02378-w
[15]	T. Kim, D. S. Kim, H. Y. Kim, J. Kwon, Some results on degenerate Daehee and Bernoulli numbers and polynomials, Adv. Diff. Equ. 2020 (2020), 311.
[16]	T. Kim, D. S. Kim, D. V. Dolgy, S. H. Lee, J. Kwon, Some identities of the higher-order type 2 Bernoulli numbers and polynomials of the second kind, CMES, 2021, DOI: 10.32604/cmes.2021.016532.
[17]	T. Kim, D. S. Kim, Degenerate zero-truncated poisson random variables, Russ. J. Math. Phys., 28 (2021), 66–72. doi: 10.1134/S1061920821010076
[18]	T. Kim, D. S. Kim, J. Kwon, H. Lee, Representations of degenerate poly-Bernoulli polynomials, J. Inequal. Appl., 2021 (2021), 58. doi: 10.1186/s13660-021-02592-0
[19]	T. Kim, D. S. Kim, L. C. Jang, H. Lee, H. Kim, Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function, Adv. Difference Equ., 2021 (2021), 175. doi: 10.1186/s13662-021-03337-5
[20]	Y. K. Ma, D. S. Kim, H. Lee, H. Kim, T. Kim, Reciprocity of poly-Dedkind-type $DC$ sums involving poly-Euler functions, Adv. Differ. Equ., 2021 (2021), 30. doi: 10.1186/s13662-020-03194-8
[21]	T. Komatsu, Hypergeometric degenerate Bernoulli polynomials and numbers, Ars Math. Contemp., 18 (2020), 163–177. doi: 10.26493/1855-3974.1907.3c2
[22]	G. Muhiuddin, W. A. Khan, U. Duran, Two variable type 2 Fubini polynomials, Mathematics, 9 (2021), 281. doi: 10.3390/math9030281
[23]	G. Muhiuddin, W. A. Khan, U. Duran, D. Al-Kadi, Some identities of the multi-poly-Bernoulli polynomials of complex variable, J. Funct. Spaces, 9 (2021), 7172054.
[24]	G. Muhiuddin, W. A. Khan, A. Muhyi, D. Al-Kadi, Some results on type 2 degenerate poly-Fubini polynomials and numbers, CMES, 2021, DOI: 10.32604/cmes.2021.016546.
[25]	C. S. Ryoo, W. A. Khan, On two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials, Mathematics, 8 (2020), 417. doi: 10.3390/math8030417
[26]	S. K. Sharma, W. A. Khan, C. S. Ryoo, A parametric kind of the degenerate Fubini numbers and polynomials, Mathematics, 8 (2020), 405. doi: 10.3390/math8030405
[27]	S. K. Sharma, W. A. Khan, S. Araci, S. S. Ahmed, New type of degenerate Daehee polynomials of the second kind, Adv. Differ. Equ., 2020 (2020), 428. doi: 10.1186/s13662-020-02891-8
[28]	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

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)