On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties
Abstract
:1. Introduction
2. The –Fibonacci–Changhee Numbers and Polynomials
3. The –Lucas–Changhee Numbers and Polynomials
4. Some Applications of the Generalized –Fibonacci–Lucas–Changhee Polynomials in Matrices
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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r | s | u | v | Sequence |
---|---|---|---|---|
0 | 1 | 1 | 1 | Fibonacci; |
2 | 1 | 1 | 1 | Lucas; |
0 | 1 | 2 | 1 | Pell; |
2 | 2 | 2 | 1 | Pell-Lucas; |
0 | 1 | k | 1 | k-Fibonacci; |
2 | k | k | 1 | k-Lucas; |
0 | 1 | 1 | 2 | Jacobsthal; |
2 | 1 | 1 | 2 | Jacobsthal-Lucas; |
Polynomial | ||||
---|---|---|---|---|
1 | 0 | 1 | Fibonacci; | |
1 | 2 | Lucas; | ||
1 | 0 | 1 | Pell; | |
1 | 2 | Pell–Lucas; | ||
1 | 0 | 1 | Jacobsthal; | |
1 | 2 | Jacobsthal–Lucas; | ||
0 | 1 | Fermat; | ||
2 | Fermat Lucas; | |||
0 | 1 | Chebyshev polynomials of the second kind; | ||
2 | Chebyshev polynomials of the first kind; |
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Zhang, C.; Khan, W.A.; Kızılateş, C. On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties. Symmetry 2023, 15, 851. https://doi.org/10.3390/sym15040851
Zhang C, Khan WA, Kızılateş C. On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties. Symmetry. 2023; 15(4):851. https://doi.org/10.3390/sym15040851
Chicago/Turabian StyleZhang, Chuanjun, Waseem Ahmad Khan, and Can Kızılateş. 2023. "On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties" Symmetry 15, no. 4: 851. https://doi.org/10.3390/sym15040851
APA StyleZhang, C., Khan, W. A., & Kızılateş, C. (2023). On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties. Symmetry, 15(4), 851. https://doi.org/10.3390/sym15040851