Fibonacci–Lucas identities and the generalized Trudi formula

Taras Goy and Mark Shattuck
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 203–217
DOI: 10.7546/nntdm.2020.26.3.203-217
Full paper (PDF, 232 Kb)

Details

Authors and affiliations

Taras Goy
Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University
57 Shevchenko St., 76018 Ivano-Frankivsk, Ukraine

Mark Shattuck
Department of Mathematics, University of Tennessee
37996 Knoxville, TN, USA

Abstract

In this paper, we evaluate determinants of several families of Hessenberg matrices having Fibonacci numbers as their nonzero entries. By the generalized Trudi formula, these determinant identities may be written equivalently as formulas for the Lucas numbers in terms of the Fibonacci. We provide both algebraic and combinatorial proofs of our determinant results. The former makes use of expansion along columns and induction, while the latter draws upon the definition of the determinant as a signed sum over the symmetric group and uses parity-changing involutions.

Keywords

• Hessenberg matrix
• Fibonacci number
• Determinant
• Trudi formula
• Lucas number

2010 Mathematics Subject Classification

• 05A19
• 11B39
• 15B05

References

1. Abd-Elhameed, W. M., & Zeyada, N. A. (2018). New identities involving generalized Fibonacci and generalized Lucas numbers, Indian J. Pure Appl. Math., 49 (3), 527–537.
2. Adegoke, K. (2018). A new Fibonacci identity and its associated summation identities, preprint, Available online at: https://arxiv.org/abs/1809.06850.
3. Azarian, M. K. (2012). Identities involving Lucas or Fibonacci and Lucas numbers as binomial sums, Int. J. Contemp. Math. Sci., 7 (45), 2221–2227.
4. Benjamin, A. T., & Cameron, N. T. (2005). Counting on determinants, Amer. Math. Monthly, 112 (6), 481–492.
5. Benjamin, A. T., Cameron, N. T., & Quinn, J. J. (2007). Fibonacci determinants – a combinatorial approach, Fibonacci Quart., 45 (1), 39–55.
6. Benjamin, A. T., & Quinn, J. J. (2003). Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America, Washington DC.
7. Benjamin, A. T., & Shattuck, M. A. (2007). Recounting determinants for a class of Hessenberg matrices, Integers, 7, #A55.
8. Bicknell-Johnson, M., & Spears, C. P. (1996). Classes of identities for the generalized Fibonacci numbers G_n = G_n−1 + G_n−c from matrices with constant valued determinants, Fibonacci Quart., 34 (2), 121–128.
9. Cereceda, J. L. (2014). Determinantal representations for generalized Fibonacci and Tribonacci numbers, Int. J. Contemp. Math. Sci., 9 (6), 269–285.
10. Cerin, Z. (2013). On factors of sums of consecutive Fibonacci and Lucas numbers, Ann. Math. Inform., 41, 19–25.
11. Civciv, H. (2008). A note on the determinant of five-diagonal matrices with Fibonacci numbers, Int. J. Contemp. Math. Sci., 3 (9), 419–424.
12. Frontczak, R. (2018). Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach, Notes Number Theory Discrete Math., 24 (2), 94–103.
13. Goy, T. (2018). On some fibinomial identities, Chebyshevskii Sb., 19 (2), 56–66 (in Russian).
14. Goy, T. (2019). On generalized Brioshci’s formula and its applications, Proceedings of the 5th Conference of the Mathematical Society of the Republic of Moldova, September 28 – October 1, 2019, Chisinau, Republic of Moldova, 70–73.
15. Goy, T., & Shattuck, M. (2019). Determinants of Toeplitz–Hessenberg matrices with generalized Fibonacci entries, Notes Number Theory Discrete Math., 25 (4), 83–95.
16. Goy, T., & Shattuck, M. (2019). Fibonacci and Lucas identities using Toeplitz-Hessenberg matrices, Appl. Appl. Math., 14 (2), 699–715.
17. Hisert, G. A. (2012). A different method for deriving Fibonacci power identities, JP J. Algebra Number Theory Appl., 24 (1), 1–26.
18. Ipek, A., & Arı, K. (2014). On Hessenberg and pentadiagonal determinants related with Fibonacci and Fibonacci-like numbers, Appl. Math. Comput., 229, 433–439.
19. Janjic, M. (2010). Hessenberg matrices and integer sequences, J. Integer Seq., 13, Article 10.7.8.
20. Kaygısız, K., & Şahin, A. (2012). Determinant and permanent of Hessenberg matrix and Fibonacci type numbers, Gen. Math. Notes, 7 (2), 32–41.
21. Keskin, R., & Demirtürk, B. (2010). Some new Fibonacci and Lucas identities by matrix methods, Internat. J. Math. Ed. Sci. Tech., 414 (3), 379–387.
22. Kılıc, E., Akkus, I., Ömür, N., & Ulutaş, Y. T. (2015). Formulas for binomial sums including powers of Fibonacci and Lucas numbers, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77 (4), 69–78.
23. Kılıc, E., & Arıkan, T. (2017). Evaluation of Hessenberg determinants via generating function approach, Filomat, 31 (15), 4945–4962.
24. Koshy, T. (2017). Fibonacci and Lucas Numbers and Applications, Wiley, New York.
25. Li, H., & MacHenry, T. (2013). Permanents and determinants, weighted isobaric polynomials, and integer sequences, J. Integer Seq., 16, Article 13.3.5.
26. Muir, T. (1960). The Theory of Determinants in the Historical Order of Development, Vol. 3, Dover Publications, New York.
27. Shattuck, M. (2008). Tiling proofs of some Fibonacci–Lucas relations, Integers, 8, #A18.
28. Tangboonduangjit, A., & Thanatipanonda, T. (2016). Determinants containing powers of generalized Fibonacci numbers, J. Integer Seq., 19, Article 16.7.1.
29. Vajda, S. (1989). Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications, John Wiley & Sons, New York.
30. Wituła, R., Słota, D., & Hetmaniok, E. (2013). Bridges between different known integer sequences, Ann. Math. Inform., 41, 255–263.
31. Zatorsky, R., & Goy, T. (2016). Permanents of triangular matrices and some general theorems on number sequences, J. Integer Seq., 19, Article 16.2.2.
32. Zatorsky, R., & Stefluk, S. (2013). On one class of partition polynomials, Algebra Discrete Math., 16 (1), 127–133.