Several expressions for moments of sums of hyperbolic secant random variables

Taekyun Kim; Dae San Kim; Taekyun Kim; Dae San Kim

doi:10.3934/era.2025244

Electronic Research Archive

2025, Volume 33, Issue 9: 5457-5470. doi: 10.3934/era.2025244

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Research note

Several expressions for moments of sums of hyperbolic secant random variables

Taekyun Kim ^{1
,
,},
Dae San Kim ²

1.
Department of Mathematics, Kwangwoon University, Seoul 139701, Korea
2.
Department of Mathematics, Sogang University, Seoul 121742, Korea

Received: 08 July 2025 Revised: 18 August 2025 Accepted: 08 September 2025 Published: 15 September 2025

In this paper, we find several expressions for the moments of the hyperbolic secant distribution and the moments of the sum of two independent distributions. They are expressed in terms of Euler and Bernoulli numbers, zeta values, an integral representation, and certain infinite series.
- hyperbolic secant random variable,
- Euler numbers,
- Bernoulli numbers
Citation: Taekyun Kim, Dae San Kim. Several expressions for moments of sums of hyperbolic secant random variables[J]. Electronic Research Archive, 2025, 33(9): 5457-5470. doi: 10.3934/era.2025244

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Abstract

In this paper, we find several expressions for the moments of the hyperbolic secant distribution and the moments of the sum of two independent distributions. They are expressed in terms of Euler and Bernoulli numbers, zeta values, an integral representation, and certain infinite series.

References

[1]	J. A. Adell, A. Lekuona, A probabilistic generalization of the Stirling numbers of the second kind, J. Number Theory, 194 (2019), 335–355. https://doi.org/10.1016/j.jnt.2018.07.003 doi: 10.1016/j.jnt.2018.07.003
[2]	M. Ha, S. Lee, Y. Seol, On degenerate Poisson random variable, Math. Comput. Model. Dyn. Syst., 30 (2024), 721–735. https://doi.org/10.1080/13873954.2024.2395808 doi: 10.1080/13873954.2024.2395808
[3]	T. Kim, D. S. Kim, Generalization of Spivey's recurrence relation, Russ. J. Math. Phys., 31 (2024), 218–226. https://doi.org/10.1134/S1061920824020079 doi: 10.1134/S1061920824020079
[4]	T. Kim, D. S. Kim, Explicit formulas for probabilistic multi-poly-Bernoulli polynomials and numbers, Russ. J. Math. Phys., 31 (2024), 450–460. https://doi.org/10.1134/S1061920824030087 doi: 10.1134/S1061920824030087
[5]	T. Kim, D. S. Kim, Probabilistic Bernoulli and Euler polynomials, Russ. J. Math. Phys., 31 (2024), 94–105. https://doi.org/10.1134/S106192084010072 doi: 10.1134/S106192084010072
[6]	T. Kim, D. S. Kim, J. Kwon, H. Lee, Probabilistic identities involving fully degenerate Bernoulli polynomials and degenerate Euler polynomials, Appl. Math. Sci. Eng., 33 (2025), 2448193. https://doi.org/10.1080/27690911.2024.2448193 doi: 10.1080/27690911.2024.2448193
[7]	L. Luo, Y. Ma, T. Kim, W. Liu, Probabilistic degenerate Laguerre polynomials with random variables, Russ. J. Math. Phys., 31 (2024), 706–712. https://doi.org/10.1134/S1061920824040095 doi: 10.1134/S1061920824040095
[8]	U. Pyo, D. V. Dolgy, The moment generating function related to Bell polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 34 (2024), 123–134. https://doi.org/10.17777/ascm2024.34.2.123 doi: 10.17777/ascm2024.34.2.123
[9]	P. Xue, Y. Ma, T. Kim, D. S. Kim, W. Zhang, Probabilistic degenerate poly-Bell polynomials associated with random variables, Math. Comput. Model. Dyn. Syst., 31 (2025), 2497367. https://doi.org/10.1080/13873954.2025.2497367 doi: 10.1080/13873954.2025.2497367
[10]	E. V. Kaplya, The generalization of the hyperbolic secant distribution and the logistic distribution in the single dostribution with variable kurtosis, (Russian) Dal'nevost. Mat. Zh., 20 (2020), 74–81. https://doi.org/10.33048/dmz2020.20.1.74
[11]	A. Yilmaz, Inverse stereographic hyperbolic secant distribution: a new symmetric circular model by rotated bilinear transformations, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 70 (2021), 871–887. https://doi.org/10.31801/cfsuasmas.906339 doi: 10.31801/cfsuasmas.906339
[12]	M. J. Fischer, Generalized Hyperbolic Secant Distributions: with Applications to Finance, Springer Science & Business Media, New York, 2013. https://doi.org/10.1007/978-3-642-45138-6
[13]	F. M. Castro-Macías, F. Pérez-Bueno, M. Vega, J. Mateos, R. Molina, A. K. Katsaggelos, Bayesian Blind Image Deconvolution using an Hyperbolic-Secant prior, in 2024 IEEE International Conference on Image Processing (ICIP), (2024), 1500–1506. https://doi.org/10.1109/ICIP51287.2024.10647808.
[14]	P. Ding, Three occurrences of the hyperbolic-secant distribution, Amer. Statist., 68 (2014), 32–35. https://doi.org/10.1080/00031305.2013.867902 doi: 10.1080/00031305.2013.867902
[15]	G. K. Smyth, A note on modelling cross-correlations: hyperbolic secant regression, Biometrika, 81 (1994), 396–402. https://doi.org/10.1093/biomet/81.2.396 doi: 10.1093/biomet/81.2.396
[16]	A. A. Shagayda, S. A. Stepin, A. G. Tarasov, Electron velocity distribution moments for collisional inhomogeneous plasma in crossed electric and magnetic fields, Russ. J. Math. Phys., 22 (2015), 532–545. https://doi.org/10.1134/S1061920815040135 doi: 10.1134/S1061920815040135
[17]	O. Barndorff-Nielsen, C. Halgreen, Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 38 (1977), 09–311. https://doi.org/10.1007/BF00533162 doi: 10.1007/BF00533162
[18]	K. Podgórski, J. Wallin, Convolution-invariant subclasses of generalized hyperbolic distributions, Comm. Statist. Theory Methods, 45 (2016), 98–103. https://doi.org/10.1080/03610926.2013.821489 doi: 10.1080/03610926.2013.821489
[19]	L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. https://doi.org/10.1007/978-94-010-2196-8
[20]	S. M. Ross, Introduction to Probability Models, Thirteenth edition, Academic Press, London, 2023. https://doi.org/10.1016/C2021-0-03471-4
[21]	D. G. Zill, Advanced Engineering Mathematics, 7th edition, Jones & Bartlett Learning, Burlington, 2022.
[22]	D. Gun, Y. Simsek, Combinatorial sums involving Stirling, Fubini, Bernoulli numbers and approximate values of Catalan numbers, Adv. Stud. Contemp. Math. (Kyungshang), 30 (2020), 503–513. http://dx.doi.org/10.17777ascm2020.30.4.503
[23]	K. S. Hwang, Note on complete convergence for weighted sums of widely negative dependent random variables under sub-linear expectations, Proc. Jangjeon Math. Soc., 28 (2025), 73–84. http://dx.doi.org/10.17777/pjms2025.28.1.73 doi: 10.17777/pjms2025.28.1.73
[24]	V. P. Maslov, T. V. Maslova, Probability theory for random variables with unboundedly growing values and its applications, Russ. J. Math. Phys., 19 (2012), 324–339. https://doi.org/10.1134/S1061920812030065 doi: 10.1134/S1061920812030065
[25]	E. I. Veleva, T. G. Ignatov, Distributions of joint sample correlation coefficients of independent normally distributed random variables, Adv. Stud. Contemp. Math. (Kyungshang), 12 (2006), 247–254. https://doi.org/10.14403/ASCM.2006.12.2.247 doi: 10.14403/ASCM.2006.12.2.247
[26]	M. S. Aydin, M. Acikgoz, S. Araci, A new construction on the degenerate Hurwitz-zeta function associated with certain applications, Proc. Jangjeon Math. Soc., 25 (2022), 195–203. http://dx.doi.org/10.17777/pjms2022.25.2.195 doi: 10.17777/pjms2022.25.2.195
[27]	D. S. Kim, T. Kim, Moment representations of fully degenerate Bernoulli and degenerate Euler polynomials, Russ. J. Math. Phys., 31 (2024), 682–690. https://doi.org/10.1134/S1061920824040071 doi: 10.1134/S1061920824040071
[28]	T. Kim, D. S. Kim, J. Kwon, H. Lee, Moments of a random variable arising from Laplacian random variable, Proc. Jangjeon Math. Soc., 27 (2024), 889–892. http://dx.doi.org/10.17777/pjms2024.27.4.889 doi: 10.17777/pjms2024.27.4.889
[29]	W. D. Baten, The probability law for the sum of $n$ independent variables, each subject to the law $(2h)^{-1}{\rm{sech}}(\pi x/2h)$, Bull. Am. Math. Soc., 40 (1934).
[30]	C. P. Dettmann, O. Georgiou, Product of $n$ independent uniform random variables, Statist. Probab. Lett., 79 (2009), 2501–2503. https://doi.org/10.1016/j.spl.2009.09.004 doi: 10.1016/j.spl.2009.09.004

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