A study on differential equations associated with \((q,h)\)-Frobenius-Euler polynomials
Authors
A. Al E'damat
- Department of Mathematics, Faculty of Science, Al-Hussein Bin Talal University, P.O. Box 20, Maan, Jordan.
W. A. Khan
- Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia.
S. A. K. Kirmani
- Department of Electrical Engineering, College of Engineering, Qassim University, Buraydah, 52571, Saudi Arabia.
U. Duran
- Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, Hatay 31200, Turkiye.
Ch.-S. Ryoo
- Department of Mathematics, Hannam University, Daejeon 34430, South Korea.
Abstract
In recent years, utilizing the generalized quantum exponential function (also known as the \((q,h)\)-exponential function) that extends and unifies the \(q\)- and \(h\)-exponential functions into a single and convenient form, \((q,h)\)-generalizations of the diverse polynomials and numbers, such as Euler and tangent polynomials and numbers, have been introduced and studied. Inspired by these studies, in this work, we focus on defining and analyzing extensions of Frobenius-Euler polynomials and numbers using the \((q,h)\)-exponential function. Also, we show that the mentioned polynomials are solutions to some higher-order differential equations. Furthermore, we examine that \((q,h)\)-Frobenius-Euler polynomials are solutions to higher-order differential equations combined with the \(q\)-Bernoulli, \(q\)-Euler, and \(q\)-Genocchi numbers and polynomials, respectively. Finally, we use a computer program to visualize the approximate roots of the mentioned polynomials.
Share and Cite
ISRP Style
A. Al E'damat, W. A. Khan, S. A. K. Kirmani, U. Duran, Ch.-S. Ryoo, A study on differential equations associated with \((q,h)\)-Frobenius-Euler polynomials, Journal of Mathematics and Computer Science, 36 (2025), no. 4, 386--398
AMA Style
Al E'damat A., Khan W. A., Kirmani S. A. K. , Duran U., Ryoo Ch.-S., A study on differential equations associated with \((q,h)\)-Frobenius-Euler polynomials. J Math Comput SCI-JM. (2025); 36(4):386--398
Chicago/Turabian Style
Al E'damat, A., Khan, W. A., Kirmani, S. A. K. , Duran, U., Ryoo, Ch.-S.. "A study on differential equations associated with \((q,h)\)-Frobenius-Euler polynomials." Journal of Mathematics and Computer Science, 36, no. 4 (2025): 386--398
Keywords
- \(q\)-numbers
- \((q,h)\)-derivative
- degenerate \(q\)-Frobenius-Euler polynomials
- differential equation
MSC
References
-
[1]
G. Bangerezako, Variational q-calculus, J. Math. Anal. Appl., 289 (2004), 650–665
-
[2]
H. B. Benaoum, (q, h)-analogue of Newton’s binomial formula. Addendum to: “h-analogue of Newton’s binomial formula”, J. Phys. A, 32 (1999), 2037–2040
-
[3]
R. D. Carmichael, The General Theory of Linear q-Difference Equations, Amer. J. Math., 34 (1912), 147–168
-
[4]
J. Cˇ erma´k, L. Nechva´tal, On (q, h)-analogue of fractional calculus, J. Nonlinear Math. Phys., 17 (2010), 51–68
-
[5]
H. F. Jackson, On q-functions and a certain difference operator, Trans. R. Soc. Edinburgh, 46 (1909), 253–281
-
[6]
H. F. Jackson, q-Difference Equations, Amer. J. Math., 32 (1910), 305–314
-
[7]
V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York (2002)
-
[8]
J.-Y. Kang, Properties of differential equations related to degenerate q-tangent numbers and polynomials, Symmetry, 15 (2023), 1–14
-
[9]
J.-Y. Kang, C.-S. Ryoo, Approximate roots and properties of differential equations for degenerate q-special polynomials, Mathematics, 11 (2023), 14 pages
-
[10]
W. A. Khan, M. S. Alatawi, U. Duran, Applications and properties for bivariate Bell-based Frobenius-type Eulerian polynomials, J. Funct. Spaces, 2023 (2023), 10 pages
-
[11]
W. A. Khan, U. Duran, J. Younis, C. S. Ryoo, On some extensions for degenerate Frobenius-Euler-Genocchi polynomials with applications in computer modeling, Appl. Math. Sci. Eng., 32 (2024), 24 pages
-
[12]
W. A. Khan, K. S. Nisar, U. Duran, M. Acikgoz, Novel results for generalized Apostol type polynomials associated with Hermite polynomials, Proc. Jangjeon Math. Soc., 26 (2023), 291–305
-
[13]
W. A. Khan, J. Younis, U. Duran, A. Iqbal, The higher-order type 2 Daehee polynomials associated with p-adic integral on Zp, Appl. Math. Sci. Eng., 30 (2022), 573–582
-
[14]
H. K. Kim, W. A. Khan, Some identities of a new type of degenerate poly-Frobenius-Euler polynomials and numbers, Proc. Jangjeon Math. Soc., 24 (2021), 33–45
-
[15]
B. Kurt, A note on the Apostol type q-Frobenius-Euler polynomials and generalizations of the Srivastava-Pinter addition theorems, Filomat, 30 (2016), 65–72
-
[16]
N. I. Mahmudov, q-analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pint´er addition theorems, Discrete Dyn. Nat. Soc., 2012 (2012), 8 pages
-
[17]
N. I. Mahmudov, On a class of q-Bernoulli and q-Euler polynomials, Adv. Difference Equ., 2013 (2013), 11 pages
-
[18]
M. R. S. Rahmat, The (q, h)-Laplace transform on discrete time scales, Comput. Math. Appl., 62 (2011), 272–281
-
[19]
B. Silindir, A. Yantir, Generalized quantum exponential function and its applications, Filomat, 33 (2019), 4907–4922
