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ON A GENERALIZATION OF GENOCCHI AND POLY-GENOCCHI POLYNOMIALS
Keywords:
poly-exponential function, probabilistic Genocchi polynomials, probabilistic poly-Genocchi polynomials, probabilistic Stirling numbers of the second kind, probabilistic type 2 poly-Bernoulli numbers, probabilistic unipoly Genocchi polynomials, Poisson random variableDOI:
https://doi.org/10.17654/0972087125017Abstract
The Genocchi numbers were introduced by the Angelo Genocchi, and have been widely studied across various fields of pure and applied mathematics.
In this paper, we define three special polynomials that are generalizations of the Genocchi polynomial and numbers, and find some relationships between the probablistic Stirling numbers of the second kind, probabilistic Bell polynomials, probablistic Bernoulli polynomials, the Stirling numbers of the first and the second kind, Genocchi polynomials and those polynomials and numbers.
Received: April 10, 2025
Revised: April 15, 2025
Accepted: May 20, 2025
References
J. A. Adell and A. Lekuona, A probabilistic generalization of the Stirling numbers of the second kind, J. Number Theory 194 (2019), 335-355.
J. A. Adell, Probabilistic Stirling numbers of the second kind and applications, J. Theoret. Probab. 35(1) (2022), 636-652.
G. E. Andrews, R. Askey and R. Roy, Special Functions (Encyclopedia of Mathematics and its Applications Book 71), Cambridge University Press, 1999.
A. Bayad and Y. Hamahata, Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math. 65(1) (2012), 15-24.
L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Co., Dordrecht, 1974.
R. Dere and Y. Simsek, Genocchi polynomials associated with the Umbral algebra, Appl. Math. Comput. 218(3) (2011), 756-761.
A. F. Horadam, Genocchi polynomials, Applications of Fibonacci Numbers, vol. 4, Winston-Salem, NC, 1990, pp. 145-166.
A. Karagenc, M. Acikgoz and S. Araci, Exploring probabilistic Bernstein polynomials: identities and applications, Appl. Math. Sci. Eng. 32(1) (2024), 2398591.
T. Kim, A note on the q-Genocchi numbers and polynomials, J. Inequal. Appl. 2007, Art. ID 71452, 8 pp.
H. K. Kim and L. C. Jang, A note on degenerate poly-Genocchi numbers and polynomials, Adv. Difference Equ. (2020), 392.
D. S. Kim and T. Kim, A note on poly-exponential and unipolyn functions, Russ. J. Math. Phys. 26 (2019), 40-49.
T. Kim and D. S. Kim, Generalization of Spivey’s recurrence relation, Russ. J. Math. Phys. 31 (2024), 218-226.
T. Kim and D. S. Kim, Probabilistic degenerate Bell polynomials associated with random variable, Russ. J. Math. Phys. 30(4) (2023), 528-542.
T. Kim and D. S. Kim, Probabilistic Bernoulli and Euler polynomials, Russ. J. Math. Phys. 31(1) (2024), 94-105.
T. Kim and D. S. Kim, Explicit formulas for probabilistic multi-poly-Bernoulli polynomials and numbers, Russ. J. Math. Phys. 31(3) (2024), 450-460.
D. S. Kim and T. Kim, Moment representations of fully degenerate Bernoulli and degenerate Euler polynomials, Russ. J. Math. Phys. 31(4) (2024), 682-690.
T. Kim and D. S. Kim, Probabilistic degenerate Bell polynomials associated with random variables, Russ. J. Math. Phys. 30(4) (2023), 528-542.
D. S. Kim and T. Kim, Probabilistic bivariate Bell polynomials, Quaestiones Mathematicae (2025), 1-14.
T. Kim, D. S. Kim, D. V. Dolgy and J. Kwon, Some identities on degenerate Genocchi and Euler numbers, Informatica 31(4) (2020), 42-51.
T. Kim, D. S. Kim, D. V. Dolgy and J. W. Park, Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials, Open Math. 19 (2021), 1588-1597.
T. Kim, D. S. Kim and J. Kwon, Probabilistic degenerate Stirling polynomials of the second kind and their applications, Math. Comput. Model. Dyn. Syst. 30(1) (2024), 16-30.
T. Kim, D. S. Kim and J. Kwon, Probabilistic identities involving fully degenerate Bernoulli polynomials and degenerate Euler polynomials, Appl. Math. Sci. Eng. 33(1) (2025), 11.
T. Kim, D. S. Kim, J. Kwon and H. Y. Kim, A note on degenerate Genocchi and poly-Genocchi numbers and polynomials, J. Inequal. Appl. (2020), 13.
S. H. Lee, L. Chen and W. Kim Probabilistic type 2 poly-Bernoulli polynomials, Eur. J. Pure Appl. Math. 17(3) (2024), 2336-2348.
Y. Ma, T. Kim and D. S. Kim, Probabilistic Lah numbers and Lah-Bell polynomials, 2024. arXiv:2406.01200v1 [math.NT].
L. Luo, T. Kim, D. S. Kim and Y. Ma, Probabilistic degenerate Bernoulli and degenerate Euler polynomials, Math. Comput. Model. Dyn. Syst. 30(1) (2024), 950-971.
L. Luo, Y. Ma, T. Kim and W. Liu, Probabilistic degenerate Laguerre polynomials with random variables, Russ. J. Math. Phys. 31(4) (2024), 706-712.
S. M. Ross, Introduction to Probability Models (Twelfth Edition), Academic Press, 2019.
Y. Simsek, Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. (2013), 87.
R. Soni, P. Vellaisamy and A. K. Pathak, Probabilistic generalization of the Bell polynomials, J. Anal. 32 (2024), 711-732.
J. Wang, Y. Ma, T. Kim and D. S. Kim, Probabilistic degenerate Bernstein polynomials, Appl. Math. Sci. Eng. 33(1) (2025), 15.
R. Xu, T. Kim, D. S. Kim and Y. Ma, Probabilistic degenerate Fubini polynomials associated with random variables, J. Nonlinear Math. Phys. 31(1) (2024), 47.
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