One Parameter Generalizations of the Fibonacci and Lucas Numbers

2006/06/29 by Mourad E. H. Ismail, Mourad E H Ismail, Ismail, Mourad E H · 1 citation
Mathematics · Physics and Astronomy · #11D25 #33C45 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #math.CA #math.CO #msc:11D25 #msc:33C45

paper · pdf · doi:10.48550/arxiv.math/0606743

arxiv created 2006/06/29 · openalex publication_date 2006/06/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give one parameter generalizations of the Fibonacci and Lucas numbers denoted by \F_n(þ)\ and \L_n(þ)\, respectively. We evaluate the Hankel determinants with entries \1/F_j+k+1(þ): 0≤ i,j ≤ n\ and \1/L_j+k+1(þ): 0≤ i,j≤ n\. We also find the entries in the inverse of \1/F_j+k+1(þ): 0≤ i,j ≤ n\ and show that all its entries are integers. Some of the identities satisfied by the Fibonacci and Lucas numbers are extended to more general numbers. All integer solutions to three diophantine equtions related to the Pell equation are also found.

Citations

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Cited by

• Fibonacci numbers, Euler's 2-periodic continued fractions and moment sequences

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