Several expressions for moments of sums of hyperbolic secant random variables

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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.

Keywords

hyperbolic secant random variable / Euler numbers / Bernoulli numbers

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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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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. Kat- saggelos, 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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[29] W. D. Baten, The probability law for the sum of n independent variables, each subject to the law (2h)⁻¹ sech(πx/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