\(h\)-analogue of Fibonacci numbers
type: Item
instance of: scholarly article
title: \(h\)-analogue of Fibonacci numbers
zbMATH Open document ID: 1359.11015
author: H. B. Benaoum
published in: The Ramanujan Journal
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0910.0038
review text: Let \[ (a)_{s:n} = \begin{cases} 1, & n = 0 \\ a(a+s) \dots a(a+(n-1)s), & n = 1,2, \dots \end{cases} \] For \(s=1\), \((a)_{1:k}=(a)_{k}\) coincides with Pochhammer's symbol. Let \[ \begin{bmatrix} n \\ k \end{bmatrix}_{h,h'} =\begin{pmatrix} n \\ k \end{pmatrix}(hh')_{n;k}. \] The \(h\)-Fibonacci number is defined for \(n > 0\) by \[ F_n^{(h,h')} = \sum_{k=0}^{\left [ \frac{n-1}{2} \right ]} \begin{bmatrix} n - 1 - k \\ k \end{bmatrix}_{h,h'}. \] Some properties of these numbers are formulated and proved, e.g. \[ F_n^{(h,h')} = _3F_1 \sum_{k=0}^{\left [ \frac{n-1}{2} \right ]} \frac{(-\frac{n}{2} + \frac{1}{2})_k(-\frac{n}{2} + 1)_k(h')_k}{(-n+1)_k}. \frac{(-4h)^k}{k!}, \] \[ F_{n+1}^{(h,h')} = F_n^{(h,h')} + hh'F_{n-1}^{(h,h'+1)}, \] \[ F_{n+2}^{(h,h')} = hh'\sum_{k=1}^n F_k^{(h,h'+1)} + 1, \] and others.
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 6358123
zbMATH Keywords: mathematical physics
zbMATH Keywords: non-commutative geometry
zbMATH Keywords: generalized Fibonacci numbers and polynomials
zbMATH Keywords: binomial coefficients
Wikidata QID: Q59492674
reviewed by: Krassimir Atanassov
MaRDI profile type: Publication
OpenAlex ID: W2089601869
arXiv ID: 0910.0038
cites work: Q5558293
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DOI: 10.1007/S11139-014-9566-7
description: scientific article; zbMATH DE number 6358123