An expression for zeta values and a summation formula via hyperbolic secant random variables

Abstract

The aim of this paper is to derive a summation formula for the series $\sum_{k=0}^{\infty} \frac{(-1)^{k}} {(2k+1)^{2n+1}}$ and an expression for $\zeta(2n+2)$ by using hyperbolic secant random variables. These identities involve Euler numbers and are obtained by computing the moments of the random variable and the moments of the sum of two independent such random variables.

Keywords

• Euler numbers; zeta function
• hyperbolic secant random variable
• moment generating function; probability density function

References

1. [1] J. A. Adell, B. Bényi, Probabilistic Stirling numbers and applications, Aequationes Math. 98, no. 6, 1627-1646, 2024.
2. [2] L. Comtet, Advanced combinatorics. The art of finite and infinite expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
3. [3] L. Holst, Probabilistic proofs of Euler identities, J. Appl. Probab. 50, no. 4, 1206-1212, 2013.
4. [4] T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal., Art. ID 581582, 11 pp., 2008.
5. [5] T. Kim, D. S. Kim, Generalization of Spivey’s recurrence relation, Russ. J. Math. Phys. 31, no. 2, 218 -226, 2024.
6. [6] T. Kim, D. S. Kim, Explicit formulas related to Euler’s product expansion for cosine function, arXiv:2407.19885.
7. [7] T. Kim, D. S. Kim, Probabilistic Bernoulli and Euler polynomials, Russ. J. Math. Phys. 31, no. 1, 94-105, 2024.
8. [8] D. S. Kim, T. Kim, Moment representations of fully degenerate Bernoulli and degenerate Euler polynomials, Russ. J. Math. Phys. 31, no. 4, 682–690, 2024.

9. [9] T. Kim, D. S. Kim, Probabilistic degenerate Bell polynomials associated with random variables, Russ. J. Math. Phys. 30, no. 4, 528–542, 2023.
10. [10] S. M. Ross, Introduction to probability models, Thirteenth edition, Academic Press, London, 2024. .
11. [11] R. Soni, P. Vellaisamy, A. K. Pathak, A probabilistic generalization of the Bell polynomials, J. Anal. 32, no. 2, 711-732, 2024.
12. [12] R. P. Stanley, Euler numbers, https://math.mit.edu > rstan.
13. [13] E. T. Whittaker, G. N.Watson, G. N. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Fourth edition, Reprinted, Cambridge University Press, New York, 1962.