Some Infinite Matrices Whose Leading Principal Minors Are Well-known Sequences
A. R. Moghaddamfar, S. Rahbariyan, S. Navid Salehy, S. Nima Salehy

TL;DR
This paper introduces new infinite matrices whose leading principal minors generate well-known integer sequences like Fibonacci, Lucas, Pell, and Jacobsthal, expanding the understanding of their structural properties.
Contribution
The paper presents novel families of infinite matrices with principal minors that produce several famous integer sequences, broadening previous scattered results.
Findings
Constructed new infinite matrices with specific sequence minors
Established links between matrix minors and well-known sequences
Extended known results to include Pell and Jacobsthal sequences
Abstract
There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of infinite matrices such that the leading principal minors of them form a famous integer (sub)sequence, such as Fibonacci, Lucas, Pell and Jacobsthal (sub)sequences.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Combinatorial Mathematics · Advanced Mathematical Identities
Some Infinite Matrices
Whose Leading Principal Minors Are Well-known Sequences
A. R. Moghaddamfar1, S. Rahbariyan1,
S. Navid Salehy2 and S. Nima Salehy2
1*Department of Mathematics,
K. N. Toosi University of Technology,
*P. O. Box -, Tehran, Iran
2*Department of Mathematics, Florida State University,
Tallahassee, FL , USA.
E-mails: [email protected] and [email protected]
In memory of Professor Michael Neumann
Abstract
There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of infinite matrices such that the leading principal minors of them form a famous integer (sub)sequence, such as Fibonacci, Lucas, Pell and Jacobsthal (sub)sequences.
footnotetext: * Mathematics Subject Classification*: 15A15, 15A23, 15B05, 15B36, 11C20.
Key words. Fibonacci sequence, Lucas sequence, Pell sequence, Jacobsthal sequence, Determinant, Toeplitz matrix, 7-matrix, Generalized Pascal triangle, Matrix factorization, Recurrence relation.
1 Notation, Definitions and Previous Results
Throughout this article, unless noted otherwise, we will use the following notation:
- •
and are two arbitrary sequences starting with .
- •
If , then where for all .
- •
is the generalized Pascal triangle associated with the sequences and (see [1]), which is introduced as follows. In fact, is a square matrix of order whose -entry obeys the following rule:
[TABLE]
and
[TABLE]
- •
is the 7-matrix associated with the sequences and of order , whose entries satisfy
[TABLE]
and
[TABLE]
This class of matrices was first introduced in [6]. In applications, the case when and is more important. In fact, we put
[TABLE]
which is called the unipotent lower triangular matrix of order . An explicit formula for -entry is also given by the following formula
[TABLE]
Moreover, we put , where signifies the transpose of matrix .
- •
is the Toeplitz matrix of order with and , , and whenever . In the case when and , we follow the notation in [7], and put . For example
[TABLE]
- •
An upper Hessenberg matrix, , is a square matrix of order where whenever and for some , , so we have
[TABLE]
- •
denotes the square matrix having 1 in the position and 0 elsewhere.
- •
(resp. ) denotes the row (resp. column ) of a matrix .
Remark. Notice that we index all matrices in this article beginning at the -th entry.
- •
is the th Fibonacci number (A000045 in [14]), which satisfies
[TABLE]
- •
is the th Lucas number (A000032 in [14]), which satisfies
[TABLE]
- •
is the th Pell number (A000129 in [14]), which satisfies
[TABLE]
- •
is the th Jacobsthal number (A001045 in [14]), which satisfies
[TABLE]
Remark. Let be integers with . The -Gibonacci sequence (or generalized Fibonacci sequence), , is recursively defined by:
[TABLE]
and
[TABLE]
The Fibonacci sequence (resp. Lucas sequence, Pell sequence, or Jacobsthal sequence) corresponds to the case (resp. , or ).
- •
Given an arbitrary sequence , we put where .
- •
To any sequence , we associate other sequences and (the binomial and inverse binomial transforms of ), which are defined through the following rules:
[TABLE]
- •
is the golden ratio and is the golden ratio conjugate.
Given an infinite matrix , denote by () the th leading principal minor of defined as the determinant of the submatrix consisting of the entries in its first rows and columns, i.e., . Here, we are interested in computing the sequence of leading principal minors , especially, in the case that is the th Fibonacci (Lucas, Pell or Jacobsthal) number. In [9], we introduced a family of matrices as follows:
[TABLE]
The matrix for instance is given by
[TABLE]
The leading principal minors now generate the Fibonacci numbers (see [9]).
In [11], we defined two other families of matrices:
[TABLE]
with
[TABLE]
and
[TABLE]
The matrices and , for instance, are shown below:
[TABLE]
and
[TABLE]
It was proved in [11] that .
Recently, in [8], it is determined all sequences and which satisfy the second-order homogeneous linear recurrence relations and for which the leading principal minors of generalized Pascal triangle , 7-matrix or Toeplitz matrix , form the Fibonacci, Lucas, Pell and Jacobsthal sequences.
In [15], using generating functions, the authors proved that any sequence can be presented in terms of a sequence of leading principal minors of an infinite matrix. In fact, according to their methods one can easily construct matrices whose determinants are famous sequences. For example, they constructed the following matrix
[TABLE]
in which the first column is the sequence of the square positive integers and the remaining part of the matrix is the truncated Pascal’s upper triangular matrix. Let denote the by upper left corner matrix of . Then , where is the th triangular number, i.e., .
In [7], in particular, it was proved that, for Toeplitz matrices
[TABLE]
and .
Table 1-3 list several other infinite matrices, whose leading principal minors form a Fibonacci or Lucas (sub)sequence. In these tables denotes .
Table 1. Some Toeplitz matrices with determinants as Fibonacci or Lucas numbers.
\begin{array}[]{|l|l|l|c|}\hline\cr\alpha&\beta&\det T_{\alpha,\beta}(n)&{\rm Refs.}\\ \hline\cr(2,1,1,\ldots)&(2,-1,0,0,\ldots)&\mathcal{F}_{2n+3}&\cite[cite]{[\@@bibref{}{cahill1}{}{}]}\\[2.84544pt] (2,1,1,\ldots)&(2,1,0,0,\ldots)&\mathcal{F}_{n+3}&\cite[cite]{[\@@bibref{}{cahill1}{}{}]}\\[2.84544pt] (1,\mathbf{i},0,0,\ldots)&(1,\mathbf{i},0,0,\ldots)&\mathcal{F}_{n+2}&\cite[cite]{[\@@bibref{}{bens, cahill1, cahill2, yang}{}{}]}\\[2.84544pt] (1,-1,0,0,\ldots)&(1,1,0,0,\ldots)&\mathcal{F}_{n+2}&\cite[cite]{[\@@bibref{}{St1}{}{}]}\\[2.84544pt] (3,1,0,0,\ldots)&(3,1,0,0,\ldots)&\mathcal{F}_{2n+4}&\cite[cite]{[\@@bibref{}{bens, St1, St2}{}{}]}\\[2.84544pt] (3,-1,0,0,\ldots)&(3,-1,0,0,\ldots)&\mathcal{F}_{2n+4}&\cite[cite]{[\@@bibref{}{St2}{}{}]}\\[2.84544pt] (2,-1,1,-1,1,\ldots)&(2,1,0,0,\ldots)&\mathcal{F}_{2n+3}&\cite[cite]{[\@@bibref{}{bens}{}{}]}\\[2.84544pt] (2,-1,1,-1,1,\ldots)&(2,-1,0,0,\ldots)&\mathcal{F}_{n+3}&\cite[cite]{[\@@bibref{}{bens}{}{}]}\\[2.84544pt] (1,\Phi,\Phi,\ldots)&(1,\phi,\phi,\ldots)&\mathcal{F}_{n+2}&\cite[cite]{[\@@bibref{}{GST}{}{}]}\\[2.84544pt] (0,-\Phi,-\Phi,\ldots)&(0,-\phi,-\phi,\ldots)&\mathcal{F}_{n}&\cite[cite]{[\@@bibref{}{GST}{}{}]}\\[2.84544pt] \hline\cr\end{array}
Table 2. Some matrices with determinants as Fibonacci or Lucas numbers.
\begin{array}[]{|l|l|l|c|}\hline\cr\alpha&\beta&{\rm Determinant}&{\rm Ref.}\\ \hline\cr(1,\mathbf{i},0,\ldots)&(1,\mathbf{i},0,\ldots)&\det\big{(}T_{\alpha,\beta}(n)+E_{11}\big{)}=\mathcal{L}_{n+1}&\cite[cite]{[\@@bibref{}{cahill1}{}{}]}\\[2.84544pt] (1,1,1,\ldots)&(1,-1,0,\ldots)&\det\big{(}T_{\alpha,\beta}(n)+\sum_{i=1}^{n}E_{ii}\big{)}=\mathcal{F}_{2n+2}&\cite[cite]{[\@@bibref{}{cahill1}{}{}]}\\[2.84544pt] (1,\mathbf{i},0,\ldots)&(1,\mathbf{i},0,\ldots)&\det\big{(}T_{\alpha,\beta}(n)+2E_{00}\big{)}=\mathcal{L}_{n+2}&\cite[cite]{[\@@bibref{}{by}{}{}]}\\[2.84544pt] \hline\cr\end{array}
Table 3. Some Pascal and 7-matrices with determinants as Fibonacci or Lucas numbers.
\begin{array}[]{|l|l|l|l|c|}\hline\cr\gamma&\alpha&\beta&{\rm Determinant}&{\rm Ref.}\\ \hline\cr 1&\alpha_{i}=\alpha_{i-1}+c&\beta_{i}=\beta_{i-1}-c^{-1}&\det P_{\alpha,\beta}(n)=\mathcal{F}_{n+2}&\cite[cite]{[\@@bibref{}{mpss}{}{}]}\\[2.84544pt] 1&\alpha_{i}=\alpha_{i-1}+1&\beta_{1}=0,\ \beta_{i}=\beta_{i+2}&\det A_{\alpha,\beta}(n)=\mathcal{F}_{n+1}&\cite[cite]{[\@@bibref{}{mpss}{}{}]}\\[2.84544pt] 1&\alpha_{i}=\alpha_{i-1}-1&\beta_{1}=0,\ \beta_{i}=\beta_{i+2}&\det A_{\alpha,\beta}(n)=\mathcal{F}_{n+1}&\cite[cite]{[\@@bibref{}{mpss}{}{}]}\\[2.84544pt] 1&\alpha_{i}=\alpha_{i-1}+\mathbf{i}&\beta_{1}=2\mathbf{i},\ \beta_{i}=\beta_{i+2}&\det A_{\alpha,\beta}(n)=\mathcal{L}_{n+1}&\cite[cite]{[\@@bibref{}{mpss}{}{}]}\\[2.84544pt] 1&\alpha_{i}=\alpha_{i-1}-\mathbf{i}&\beta_{1}=-2\mathbf{i},\ \beta_{i}=\beta_{i+2}&\det A_{\alpha,\beta}(n)=\mathcal{L}_{n+1}&\cite[cite]{[\@@bibref{}{mpss}{}{}]}\\[2.84544pt] \hline\cr\end{array}
The purpose of this article is to find some new infinite families of matrices whose leading principal minors form a subsequence of the sequences , , and .
2 Preliminaries
In this section, we prove several auxiliary results to be used later. First of all, we introduce the notion of equimodular matrices (see [16]).
Definition 1
Infinite matrices and are equimodular if their leading principal minors are the same, that is
[TABLE]
Lemma 1** (Corollary 2.6, [8])**
Let and be two sequences starting with a common first term. Then, the following matrices are equimodular:
[TABLE]
The following lemma, which is taken from [7], determines recursive and explicit formula for the leading principal minors for special classes of Toeplitz matrices.
Lemma 2
([7]) Let . For each positive integer , let . Then , and for ,
[TABLE]
Moreover, an explicit formula for the case when , is as follows:
[TABLE]
Lemma 3
Let . Let with and for each , . Then there hold:
* for each , .*
* and for each , \hat{\alpha}_{i}=\left\{\begin{array}[]{ll}a-c&i\ \mbox{is odd,}\\[5.69046pt] c-a&i\ \mbox{is even.}\\ \end{array}\right.*
Proof. The following easy computations
[TABLE]
and
[TABLE]
conclude the results.
Unless noted otherwise, we use the following notation in this section:
- •
, where and are integers with ,
- •
, where is an integer. Moreover, we assume that .
Lemma 4
The sequences and satisfy linear recursions of order . More precisely, we have
[TABLE]
and
[TABLE]
Proof. By the definition, we have the following recurrence relation
[TABLE]
which can be written as
[TABLE]
Its characteristic equation is
[TABLE]
and we denote by and the characteristic roots. Now, the general solution of Eq. (3) is given by
[TABLE]
where and are constants. In order to verify Eq. (1), we first notice that, for every :
[TABLE]
Therefore, we verify Eq. (1) by a direct calculation. Clearly and . Let us for the moment assume that . Then, we have
[TABLE]
as required.
The second part (i.e., the equalities in Eq. (2)) is similar and we leave it to the reader.
Lemma 5
If , then .
Proof. We will proceed by induction with respect to . The case is trivial:
[TABLE]
Assume the identity is true when , that is,
[TABLE]
We now prove that it is also true for . Easy calculations show that
[TABLE]
which completes the proof.
Lemma 6
If , then
[TABLE]
and
[TABLE]
Proof. If or , then the proof is straightforward. Therefore, we may assume that . Put
[TABLE]
We claim that
[TABLE]
To prove this we will proceed by induction with respect to . It is clear for . Suppose that the claim is true for . Our task is to show that it is also true for . Again, we verify the result by a direct calculation:
[TABLE]
as claimed.
Now, we have , that is
[TABLE]
or equivalently
[TABLE]
Comparing -entries on both sides of this equation yields
[TABLE]
which is the Eq. (6).
Moreover, since , , or equivalently
[TABLE]
and this immediately implies the following:
[TABLE]
which is the Eq. (7). This completes the proof of the lemma.
An immediate consequence of Lemma 6 is the following, which contains a list of well-known identities.
Corollary 1
If , then the following identities hold:
,
, (Cassini’s identity)
,
. (Simpson’s identity)
3 Main Results
As we mentioned in the Introduction, we are going to obtain some new matrices whose leading principal minors form the Fibonacci, Lucas, Pell and Jacobsthal sequences. We start with the following theorem.
Theorem 1
If , where and are integers with , then for any nonnegative integer there holds
[TABLE]
Proof. For , we have nothing to do. Therefore, we may assume that . We use the matrix factorization method. Let denote the matrix . We claim that
[TABLE]
where with
[TABLE]
(we recall that ) and with
[TABLE]
The matrix is a lower triangular matrix with 1’s on the diagonal, whereas is an upper Hessenberg matrix. The matrices and for instance are given by
[TABLE]
[TABLE]
In what follows, for convenience, we will let , and . For the proof of the claimed factorization Eq. (8) we compute the -entry of , that is
[TABLE]
As a matter of fact, we should establish
[TABLE]
and
[TABLE]
Let us do the required calculations. First, suppose that . Then
[TABLE]
which shows that .
Next, assume that . In this case, using Lemma 5, we obtain
[TABLE]
and hence we have .
Finally, we must establish Eq. (11). To do this, we assume that . We distinguish two cases separately: and .
Case 1. . We calculate the sum in question:
[TABLE]
as required.
Case 2. . We proceed analogously. In this case, we have
[TABLE]
as required.
Returning back to the Eq. (8), we conclude that
[TABLE]
Now, expanding the determinant of along the rows , one can easily get
[TABLE]
This completes the proof of the theorem.
Although we have restricted ourselves to integral values in Theorem 1, it holds for arbitrary values of all parameters , and . In the sequel, we will consider some applications of Theorem 1 in some simple, but interesting, cases.
Corollary 2
Let be an integer and . Then there holds
[TABLE]
In particular, we have
If , then .
If , then .
Proof. Let . Using Theorem 1 and Corollary 1 , we have
[TABLE]
The rest follows immediately.
Corollary 3
Let be an integer and . Then for every there holds
[TABLE]
In particular, for we conclude that .
Proof. Let . Again, using Theorem 1 and Corollary 1 , we have
[TABLE]
The rest follows immediately.
Notice that, using Lemma 4, if , then
[TABLE]
Corollary 4
*If , then the following infinite integer matrices:
\begin{array}[]{ll}T_{\tilde{\alpha},\alpha}=\left(\begin{array}[]{ccccc}1&1&2&3&\cdot\\ -1&1&1&2&\cdot\\ 2&-1&1&1&\cdot\\ -3&2&-1&1&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ \end{array}\right),&T_{\alpha,\tilde{\alpha}}=\left(\begin{array}[]{ccccc}1&-1&2&-3&\cdot\\ 1&1&-1&2&\cdot\\ 2&1&1&-1&\cdot\\ 3&2&1&1&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ \end{array}\right),\\[42.67912pt] P_{\check{\tilde{\alpha}},\check{\alpha}}=\left(\begin{array}[]{ccccc}1&2&5&13&\cdot\\ 0&2&7&20&\cdot\\ 1&3&10&30&\cdot\\ 1&4&14&44&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ \end{array}\right),&P_{\check{\alpha},\check{\tilde{\alpha}}}=\left(\begin{array}[]{ccccc}1&0&1&1&\cdot\\ 2&2&3&4&\cdot\\ 5&7&10&14&\cdot\\ 13&20&30&44&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ \end{array}\right),\\[42.67912pt] A_{\check{\tilde{\alpha}},\alpha}=\left(\begin{array}[]{ccccc}1&1&2&3&\cdot\\ 0&2&3&5&\cdot\\ 1&2&5&8&\cdot\\ 1&3&7&13&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ \end{array}\right),&A_{\check{\alpha},\tilde{\alpha}}=\left(\begin{array}[]{cccccc}1&-1&2&-3&\cdot\\ 2&0&1&-1&\cdot\\ 5&2&1&0&\cdot\\ 13&7&3&1&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot\\ \end{array}\right),\\[42.67912pt] \end{array}**
are equimodular. In precisely, we have
[TABLE]
for all .
Proof. Follows directly from Corollary 2 and Lemmas 1 and 4.
Corollary 5
Let and be arbitrary complex numbers, and for each positive integer , let . Then , and for ,
[TABLE]
if and only if
[TABLE]
Furthermore, if Eq. holds, then we have . In particular, there hold
* If , then ,*
* If , then ,*
* If , then ,*
* If , then ,*
* If , then ,*
* If , then .*
Proof. The sufficiency is clear. To prove the necessity, using Lemma 2 we have
[TABLE]
and by comparing this with Eq. (13), we then obtain the system of equalities
[TABLE]
which has two solutions of the forms
[TABLE]
The rest of the proof is obvious.
Corollary 6
Let and be arbitrary complex numbers, and for each positive integer , let . Then , and for ,
[TABLE]
if and only if
[TABLE]
Furthermore, if Eq. holds, then we have . In particular, there hold
* If , then ,*
* If , then .*
Proof. The sufficiency is clear. To prove the necessity, using Lemma 2 we have
[TABLE]
and by comparing this with Eq. (15), we then obtain the system of equalities
[TABLE]
which has two solutions of the forms
[TABLE]
The rest of the proof is obvious.
Corollary 7
Let and be arbitrary complex numbers, and for each positive integer , let . Then , and for ,
[TABLE]
if and only if
[TABLE]
Furthermore, if Eq. holds, then we have . In particular, we have
* If , then .*
* If , then .*
Proof. The sufficiency is clear. To prove the necessity, using Lemma 2 we have
[TABLE]
and by comparing this with Eq. (17), we then obtain the system of equalities
[TABLE]
which has two solutions of the forms
[TABLE]
The rest of the proof is obvious.
Corollary 8
Let and be two sequences starting with a common first term. Then there hold:
* If and , then ,*
* If and , then .*
Proof. Both statements follow directly from Lemma 1, Lemma 3 and Corollary 7.
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