Discrete quasiprobability distributions involving Bernoulli polynomials

Bander Almutairi; Bander Almutairi

doi:10.3934/math.2023645

AIMS Mathematics

2023, Volume 8, Issue 6: 12819-12829. doi: 10.3934/math.2023645

Previous Article Next Article

Research article

Discrete quasiprobability distributions involving Bernoulli polynomials

Bander Almutairi ^,

Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 20 December 2022 Revised: 27 February 2023 Accepted: 06 March 2023 Published: 30 March 2023
MSC : 60E05, 62E15

The aim of this short paper is to present a new family of discrete densities with two parameters based on Bernoulli numbers and polynomials. We use the properties of such numbers in order to compute the first moments and the density of a finite sum of such independent variables.
- quasiprobability,
- Bernoulli numbers,
- Stirling numbers,
- Planck's law,
- poisson distribution
Citation: Bander Almutairi. Discrete quasiprobability distributions involving Bernoulli polynomials[J]. AIMS Mathematics, 2023, 8(6): 12819-12829. doi: 10.3934/math.2023645

Related Papers:

Abstract

The aim of this short paper is to present a new family of discrete densities with two parameters based on Bernoulli numbers and polynomials. We use the properties of such numbers in order to compute the first moments and the density of a finite sum of such independent variables.

References

[1]	P. Billingsley, Probability and measure, New York: John Wiley and Sons, 1995.
[2]	R. P. Brent, D. Harvey, Fast computation of Bernoulli, Tangent and Secants numbers, Comput. Anal. Math., 50 (2013), 127–142.
[3]	L. C. Jang, D. S. Kim, T. Kim, H. Lee, Some identities involving derangement polynomials and numbers and moments of Gamma random variables, J. Funct. Spaces, 2020 (2020), 6624006. https://doi.org/10.1155/2020/6624006 doi: 10.1155/2020/6624006
[4]	K. Kamano, Sums of product of hypergeometric Bernoulli numbers, J. Number Theory, 130 (2010), 2259–2271. http://dx.doi.org/10.1016/j.jnt.2010.04.005 doi: 10.1016/j.jnt.2010.04.005
[5]	T. Kim, Y. Yao, D. S. Kim, H. I. Kwon, Some identities involving special numbers and moments of random variables, Rocky Moutain J. Math., 49 (2019), 521–538. http://dx.doi.org/10.1216/RMJ-2019-49-2-521 doi: 10.1216/RMJ-2019-49-2-521
[6]	T. Kim, D. S. Kim, H. Lee, S. H. Park, Dimorphic properties of Bernoulli random variable, Filomat, 36 (2022), 1711–1717.
[7]	T. Kim, D. S. Kim, Degenerate zero-truncated Poisson random variables, Russ. J. Math. Phys., 28 (2021), 66–72. https://doi.org/10.1134/S1061920821010076 doi: 10.1134/S1061920821010076
[8]	T. Kim, D. S. Kim, H. K. Kim, Poisson degenerate central moments related to degenerate Dowling and degenerate r-Dowling polynomials, Appl. Math. Sci. Eng., 30 (2022), 583–597. https://doi.org/10.1080/27690911.2022.2118736 doi: 10.1080/27690911.2022.2118736
[9]	T. Kim, D. S. Kim, D. V. Dolgy, J. W. Park, Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials, Open Math., 19 (2021), 1588–1597. https://doi.org/10.1515/math-2021-0116 doi: 10.1515/math-2021-0116
[10]	T. Kim, D. S. Kim, Some identities on Degenerate r-Striling numbers via Boson operators, Russ. J. Math. Phys., 29 (2022), 508–517.
[11]	T. Kim, D. S. Kim, L. C. Jang, H. Y. Kim, A note on discrete degenerate random variables, Proc. Jangjeon Math. Soc., 23 (2020), 125–135.
[12]	F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, NIST handbook of mathematical functions, Cambridge: Cambridge University Press, 2010.
[13]	H. S. Vandiver, An arithmetical theory of the Bernoulli numbers, Trans. Amer. Math. Soc., 51, (1942), 502–531.
[14]	E. Zeidler, Quantum field theory: a bridge between mathematicians and physicists, Berlin: Springer, 2009.

Reader Comments

Your name:*

Email:*
© 2023 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)