Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912553841459200 |
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| author | Mane, S. R. |
| author_facet | Mane, S. R. |
| contents | We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are presented, using Fuss-Catalan numbers. Properties of the roots are enumerated. We quantify the accuracy of asymptotic approximations for $F_k(n)$ for $n\gg1$. Our results subsume and extend some results published by previous authors. We also present a basis (or `fundamental solutions') to solve the above recurrence for arbitrary initial conditions. We comment on the use of generating functions and multinomial sums for the $k$-generalized Fibonacci numbers and related sequences. We note that the resulting multinomial sums are Dickson polynomials of the second kind in several variables. We also present what may be a new identity for companion matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers Mane, S. R. Combinatorics Number Theory 11B39, 11B37, 39A06 We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are presented, using Fuss-Catalan numbers. Properties of the roots are enumerated. We quantify the accuracy of asymptotic approximations for $F_k(n)$ for $n\gg1$. Our results subsume and extend some results published by previous authors. We also present a basis (or `fundamental solutions') to solve the above recurrence for arbitrary initial conditions. We comment on the use of generating functions and multinomial sums for the $k$-generalized Fibonacci numbers and related sequences. We note that the resulting multinomial sums are Dickson polynomials of the second kind in several variables. We also present what may be a new identity for companion matrices. |
| title | Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers |
| topic | Combinatorics Number Theory 11B39, 11B37, 39A06 |
| url | https://arxiv.org/abs/2410.07922 |