Expressing the generalized Fibonacci polynomials in terms of a tridiagonal determinant. (English) Zbl 1432.11014
Summary: In the note, the authors express the generalized Fibonacci polynomials in terms of a tridiagonal determinant. Consequently, they also express the Fibonacci numbers and polynomials in terms of tridiagonal determinants.
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11B83 | Special sequences and polynomials |
| 11C20 | Matrices, determinants in number theory |
| 11Y55 | Calculation of integer sequences |
Keywords:
generalized Fibonacci polynomials; Fibonacci number; Fibonacci polynomials; tridiagonal determinantReferences:
| [1] | T. Amdeberhan, X. Chen, V. H. Moll, and B. E. Sagan, Generalized Fibonacci polynomials and Fibonomial coefficients, Ann. Comb. 18 (2014) 541–562; Available online at http://dx.doi.org/10.1007/s00026-014-0242-9. · Zbl 1375.05010 |
| [2] | N. Bourbaki, Elements of Mathematics: Functions of a Real Variable: Elementary Theory, Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004; Available online at http: //dx.doi.org/10.1007/978-3-642-59315-4. 174FENG QI - BAI-NI GUO |
| [3] | R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing Co., Inc., River Edge, NJ, 1997; Available online at http://dx.doi.org/ 10.1142/9789812386304. · Zbl 0968.11013 |
| [4] | V. Higgins and C. Johnson, Inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices, Linear Algebra Appl. 489 (2016), 104–122; Available online at http://dx.doi.org/10.1016/j.laa.2015.10. 004. · Zbl 1325.15032 |
| [5] | R. S. Martin and J. H. Wilkinson, Handbook Series Linear Algebra: Similarity reduction of a general matrix to Hessenberg form, Numer. Math. 12 (1968), no. 5, 349–368; Available online at http://dx.doi.org/10.1007/BF02161358. · Zbl 0184.37504 |
| [6] | F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268 (2015), 844–858; Available online at http://dx.doi.org/10.1016/j. amc.2015.06.123. · Zbl 1410.11018 |
| [7] | F. Qi and R. J. Chapman, Two closed forms for the Bernoulli polynomials, J. Number Theory 159 (2016), 89–100; Available online at http://dx.doi.org/10. 1016/j.jnt.2015.07.021. · Zbl 1400.11070 |
| [8] | A. Sahin and K. Kaygisiz, Determinantal and permanental representation of generalized Fibonacci polynomials, arXiv preprint (2011), available online at http: //arxiv.org/abs/1111.4066. |
| [9] | C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 2015:219, 8 pages; Available online at http://dx.doi.org/10. 1186/s13660-015-0738-9. · Zbl 1366.11052 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
