The Filbert Matrix
Abstract
A Filbert matrix is a matrix whose (i,j) entry is 1/F_(i+j1), 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1999
 arXiv:
 arXiv:math/9905079
 Bibcode:
 1999math......5079R
 Keywords:

 Rings and Algebras;
 Combinatorics;
 11B39;
 11B65;
 15A09
 EPrint:
 9 pages