The Filbert Matrix

1999/05/12 by Thomas M. Richardson
Mathematics · #math.RA #math.CO #msc:11B39 #msc:11B65 #msc:15A09

paper · pdf

published as Fibonacci Quart. 39 (2001), no. 3, 268--275. · 9 pages; submitted to The Fibonacci Quarterly

arxiv created 1999/05/12 · arxiv updated 2009/11/30

Abstract

A Filbert matrix is a matrix whose (i,j) entry is 1/F_(i+j-1), where F_n is the nth Fibonacci number. The inverse of the n by n Filbert matrix resembles the inverse of the n by n Hilbert matrix, and we prove that it shares the property of having integer entries. We prove that the matrix formed by replacing the Fibonacci numbers with the Fibonacci polynomials has entries which are integer polynomials. We also prove that certain Hankel matrices of reciprocals of binomial coefficients have integer entries, and we conjecture that the corresponding matrices based on Fibonomial coefficients have integer entries. Our method is to give explicit formulae for the inverses.

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