Abstract
Let \(Y\) be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of Bernoulli polynomials and Euler polynomials, namely the probabilistic Bernoulli polynomials associated \(Y\) and the probabilistic Euler polynomials associated with \(Y\). Also, we introduce the probabilistic \(r\)-Stirling numbers of the second associated \(Y\), the probabilistic two variable Fubini polynomials associated \(Y\), and the probabilistic poly-Bernoulli polynomials associated with \(Y\). We obtain some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of \(Y\), we treat the gamma random variable with parameters \(\alpha,\beta > 0\), the Poisson random variable with parameter \(\alpha >0\), and the Bernoulli random variable with probability of success \(p\).
DOI 10.1134/S106192084010072
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, U. S. Government Printing Office, Washington, DC, 1964.
C. R. Adams and A. P. Morse, “Random Sampling in the Evaluation of a Lebesgue Integral”, Bull. Amer. Math. Soc., 45:6 (1939), 442–447.
J. A. Adell, “Probabilistic Stirling Numbers of the Second Kind and Applications”, J. Theoret. Probab., 35:1 (2022), 636–652.
A. Bagdasaryan, S. Araci, M. Acikgoz, and Y. He, “Some New Identities on the Apostol-Bernoulli Polynomials of Higher Order Derived from Bernoulli Basis”, J. Nonlinear Sci. Appl., 9:5 (2016), 2697–2704.
L. Carlitz, “Some Polynomials Related to the Bernoulli and Euler Polynomials”, Util. Math., 19 (1981), 81–127.
S.-K. Chung, G.-W. Jang, J. Kwon, and J. Lee, “Some Identities of the Degenerate Changhee Numbers of Second Kind Arising from Differential Equations”, Adv. Stud. Contemp. Math. (Kyungshang), 28:4 (2018), 577–587.
L. Comtet, Advanced Combinatorics, The art of finite and infinite expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
D. Gun and Y. Simsek, “Combinatorial Sums Involving Stirling, Fubini, Bernoulli Numbers and Approximate Values of Catalan Numbers”, Adv. Stud. Contemp. Math. (Kyungshang), 30:4 (2020), 503–513.
M. Kaneko, “Poly–Bernoulli Numbers”, J. Théor. Nombres Bordeaux, 9:1 (1997), 221–228.
D. Kim and T. Kim, “A Note on Poly-Bernoulli and Higher-Order Poly-Bernoulli Polynomials”, Russ. J. Math. Phys., 22:1 (2015), 26–33.
D. S. Kim and T. Kim, “Normal Ordering Associated with \(\lambda\)-Whitney Numbers of the First Kind in \(\lambda\)-Shift Algebra”, Russ. J. Math. Phys., 30:3 (2023), 310–319.
D. S. Kim and T. Kim, “A Note on a New Type of Degenerate Bernoulli Numbers”, Russ. J. Math. Phys., 27:2 (2020), 227–235.
T. Kim and D. S. Kim, “Probabilistic Degenerate Bell Polynomials Associated with Random Variables”, Russ. J. Math. Phys., 30:4 (2023), 528–542.
T. Kim and D. S. Kim, “Some Identities Involving Degenerate Stirling Numbers Associated with Several Degenerate Polynomials and Numbers”, Russ. J. Math. Phys., 30:1 (2023), 62–75.
T. Kim and D. S. Kim, “Some Results on Degenerate Fubini and Degenerate Bell Polynomials”, Appl. Anal. Discrete Math., 17:2 (2023), 548–560.
T. Kim and D. S. Kim, “Combinatorial Identities Involving Degenerate Harmonic and Hyperharmonic Numbers”, Adv. in Appl. Math., 148:102535 (2023), 15.
T. Kim, D. S. Kim, and H. K. Kim, “Some Identities Involving Bernoulli, Euler and Degenerate Bernoulli Numbers and Their Applications”, Appl. Math. Sci. Eng., 31:1 (2023), 12.
T. Kim, D. S. Kim, and J. Kwon, “Some Identities Related to Degenerate \(r\)-Bell and Degenerate Fubini Polynomials”, Appl. Math. Sci. Eng., 31:1 (2023), 13.
B. F. Kimball, “A Generalization of the Bernoulli Polynomial of Order One”, Bull. Amer. Math. Soc., 41:12 (1935), 894–900.
A. Leon-Garcia, Probability and Random Processes for Electronic Engineering, Addison-Wesley series in electrical and computer engineering, Addison-Wesley, 1994.
J.-W. Park, B. M. Kim, and J. Kwon, “Some Identities of the Degenerate Bernoulli Polynomials of the Second Kind Arising from \(\lambda\)-Sheffer Sequences”, Proc. Jangjeon Math. Soc., 24:3 (2021), 323–342.
J.-W. Park and S.-H. Rim, “On the Modified \(q\)-Bernoulli Polynomials with Weight”, Proc. Jangjeon Math. Soc., 17:2 (2014), 231–236.
S. Roman, The Umbral Calculus, New York, 1984 x+193 pp. ISBN: 0-12-594380-6.
S. M. Ross, Introduction to Probability Models, Twelfth edition of Academic Press, London, 2019.
K. Shiratani, “Kummer’s Congruence for Generalized Bernoulli Numbers and Its Application”, Mem. Fac. Sci. Kyushu Univ. Ser. A, 26 (1972), 119–138.
R. Soni, A. K. Pathak, and P. Vellaisamy, A Probabilistic Extension of the Fubini Polynomials.
B. Q. Ta, “Probabilistic Approach to Appell Polynomials”, Expo. Math., 33:3 (2015), 269–294.
H. Teicher, “An Inequality on Poisson Probabilities”, Ann. Math. Statist., 26 (1955), 147–149.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Publisher’s note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kim, T., Kim, D.S. Probabilistic Bernoulli and Euler Polynomials. Russ. J. Math. Phys. 31, 94–105 (2024). https://doi.org/10.1134/S106192084010072
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S106192084010072