Insight into degenerate Bell-based Bernoulli polynomials with applications
Authors
U. Duran
- Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkey.
S. Araci
- Department of Computer Engineering, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Turkey.
M. Acikgoz
- Department of Mathematics, Faculty of Arts and Science, Gaziantep University, TR-27010 Gaziantep, Turkey.
Abstract
Recently, the Bell-based Stirling polynomials of the second kind and the
Bell-based Bernoulli polynomials [{U. Duran, S. Araci, M. Acikgoz, Axioms, \textbf{10} (2021), 23 pages}] have been considered, and some of their
properties and applications in umbral calculus have been derived and
analyzed. In this work, a degenerate form of the Bell-based Stirling
polynomials of the second kind is defined, and several fundamental
properties and formulas for these polynomials are investigated and presented
in detail. Then, a degenerate form of the Bell-based Bernoulli polynomials
of order \(\alpha \) is defined and a plenty of their properties\ are examined
in different aspects. Several correlations with other polynomials and
numbers in literature, symmetric identities, implicit summation formulas,
derivative properties and addition formulas for the mentioned new
polynomials are derived in detail, and some special cases of these results
are investigated. Also, the degenerate Bell-based Bernoulli polynomials of
order \(\varepsilon \) are studied in \(\lambda \)-umbral calculus and
interesting relations and formulas are developed. Furthermore, the
application of \(\lambda \)-umbral calculus to Bell-based degenerate Bernoulli
polynomials of order \(\varepsilon \) shows a correlation with higher-order
degenerate derangement polynomials. Finally, a representation of the
degenerate differential operator on the degenerate Bell-based Bernoulli
polynomials of order \(\varepsilon \) is provided.
Share and Cite
ISRP Style
U. Duran, S. Araci, M. Acikgoz, Insight into degenerate Bell-based Bernoulli polynomials with applications, Journal of Mathematics and Computer Science, 41 (2026), no. 2, 264--283
AMA Style
Duran U., Araci S., Acikgoz M., Insight into degenerate Bell-based Bernoulli polynomials with applications. J Math Comput SCI-JM. (2026); 41(2):264--283
Chicago/Turabian Style
Duran, U., Araci, S., Acikgoz, M.. "Insight into degenerate Bell-based Bernoulli polynomials with applications." Journal of Mathematics and Computer Science, 41, no. 2 (2026): 264--283
Keywords
- Bernoulli polynomials
- Bell polynomials
- mixed-type polynomials
- Stirling numbers of the second kind
- degenerate exponential function
- umbral calculus
MSC
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