MathDB

Fibonomial reciprocal Hankel inverse formula conjecture

From papers

Let FnF_n be the nnth Fibonacci number, let e(n,i,k)e(n,i,k) be the exponent used to determine the sign in the source, and let dk(r)d_k(r) be the Fibonomial coefficient corresponding to (k+r−1r)\binom{k+r-1}{r}. For positive integers nn and rr, define the n×nn\times n matrix D(n,r)D(n,r) by

Dij=Dij(n,r)=∑k=0j−1(−1)e(n,i,k)((n+i+r−2i))((ni))((n+k+r−2k))((nk))Fi2∏l=0r−3Fi+j+lFr∏l=0r−2Fi+k+l.D_{ij}=D_{ij}(n,r)=\sum_{k=0}^{j-1}(-1)^{e(n,i,k)}\Biggl(\binom{n+i+r-2}{i}\Biggr)\Biggl(\binom{n}{i}\Biggr)\Biggl(\binom{n+k+r-2}{k}\Biggr)\Biggl(\binom{n}{k}\Biggr)\frac{F_i^2\prod_{l=0}^{r-3}F_{i+j+l}}{F_r\prod_{l=0}^{r-2}F_{i+k+l}}.

Here the parenthesized binomial symbols are Fibonomial coefficients. Fibonomial reciprocal Hankel inverse conjecture. The matrix D(n,r)D(n,r) is the inverse of Rn(dk)R_n(d_k). The surrounding discussion presents this as a proposed formula for the inverse of reciprocal Hankel matrices based on Fibonomial coefficients; the source does not establish the formula in the supplied passage.

Progress summary

Open

The formula remains an unproved conjecture: only finite computer checks are reported, and no verified proof or disproof has been found.

A 1999 paper states Conjecture 6.1, asserting that the explicitly defined matrix D(n,r)D(n,r) is the inverse of the reciprocal Hankel matrix Rn(dk)R_n(d_k) built from Fibonomial coefficients. The paper presents this as an analogue of a proved binomial-coefficient formula.

Known results

  • The conjecture was computationally verified for n≤16n \le 16 and r≤10r \le 10.
  • The corresponding reciprocal Hankel inverse formula for ordinary binomial coefficients is proved; the Fibonomial formula is offered as its analogue.
  • The same paper relates integrality of the Fibonomial inverses to integrality of the corresponding binomial-coefficient matrices.

Current status (as of August 2026): The conjecture remains open; finite computational verification is recorded, but no publicly retrieved proof, disproof, or independently verified correction has been found.

Sources
Sources & referencesView supporting material

Primary source

Thomas M. Richardson, “The Filbert Matrix”, arXiv:math/9905079 (1999).

fibonacci-numbershankel-matricesinteger-matrices

Solutions 1

Counterexample

Counterexample to the formula as printed

Take n=1n=1 and r=3r=3. Then

d1(3)=(33)F=1,d_1(3)=\binom{3}{3}_F=1,

so R1(dk)=[1]R_1(d_k)=[1], whose inverse is also [1][1].

For the proposed matrix D(1,3)D(1,3), set i=j=1i=j=1. The defining sum contains only the term k=0k=0, and the printed exponent gives

e(1,1,0)=1(1+0+1)+(12)+(02)+1=3.e(1,1,0) = 1(1+0+1)+\binom{1}{2}+\binom{0}{2}+1 =3.

The unsigned part of the summand is

(31)F(11)F(20)F(10)FF12F2F3F1F2=2⋅1⋅1⋅1⋅12=1.\binom31_F\binom11_F\binom20_F\binom10_F \frac{F_1^2F_2}{F_3F_1F_2} = 2\cdot1\cdot1\cdot1\cdot\frac12 =1.

Hence

D(1,3)=[−1],D(1,3)R1(dk)=[−1]≠[1].D(1,3)=[-1], \qquad D(1,3)R_1(d_k)=[-1]\ne[1].

Therefore the displayed formula for D(n,r)D(n,r) is false as printed.

Scope and likely explanation

This refutes only the displayed formula. I checked both the 1999 arXiv version and the published 2001 Fibonacci Quarterly version; both print the same exponent and formula. Thus this is not a MathDB transcription error.

The paper nevertheless reports computational verification for n≤16n\le16 and r≤10r\le10, which includes the example above. The most likely explanation is therefore a typographical sign discrepancy between the displayed formula and the formula used in those computations—not a failure of the intended conjecture.

Multiplying column jj by (−1)n+j+1(-1)^{n+j+1} repairs every case checked exactly for 1≤n≤81\le n\le8 and 2≤r≤102\le r\le10. I do not claim an all-index proof of this corrected formula.

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Samuel Schlesinger · · edited