vol18_pp564-588. 215KB Jun 04 2011 12:06:04 AM

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 564-588, August 2009

ELA

http://math.technion.ac.il/iic/ela

GENERALIZED PASCAL TRIANGLES AND TOEPLITZ MATRICES∗
A. R. MOGHADDAMFAR† AND S. M. H. POOYA‡

Abstract. The purpose of this article is to study determinants of matrices which are known as
generalized Pascal triangles (see R. Bacher. Determinants of matrices related to the Pascal triangle.
J. Th´
eor. Nombres Bordeaux, 14:19–41, 2002). This article presents a factorization by expressing
such a matrix as a product of a unipotent lower triangular matrix, a Toeplitz matrix, and a unipotent
upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant
of a Toeplitz matrix. This equality allows for the evaluation of a few determinants of generalized
Pascal matrices associated with certain sequences. In particular, families of quasi-Pascal matrices
are obtained whose leading principal minors generate any arbitrary linear subsequences (Fnr+s )n≥1
or (Lnr+s )n≥1 of the Fibonacci or Lucas sequence. New matrices are constructed whose entries are

given by certain linear non-homogeneous recurrence relations, and the leading principal minors of
which form the Fibonacci sequence.

Key words. Determinant, Matrix factorization, Generalized Pascal triangle, Generalized symmetric (skymmetric) Pascal triangle, Toeplitz matrix, Recursive relation, Fibonacci (Lucas, Catalan)
sequence, Golden ratio.

AMS subject classifications. 15A15, 11C20.

1. Introduction. Let P (∞) be the infinite symmetric matrix with entries Pi,j =

i+j 
for i, j ≥ 0. The matrix P (∞) is hence the famous Pascal triangle yielding the
i
binomial coefficients. The entries of P (∞) satisfy the recurrence relation Pi,j =
Pi−1,j + Pi,j−1 . Indeed, this matrix has the following form:







P (∞) = 





1
1
1
1
1
..
.

1
2
3
4
5
..

.

1 1 1 ...
3 4 5 ...
6 10 15 . . .
10 20 35 . . .
15 35 70 . . .
..
..
.. . .
.
.
.
.













∗ Received by the editors January 19, 2009. Accepted for publication August 1, 2009. Handling
Editor: Michael J. Tsatsomeros.
† Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology, P.
O. Box 16315-1618, Tehran, Iran (moghadam@kntu.ac.ir). School of Mathematics, Institute
for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran
(moghadam@mail.ipm.ir). This research was in part supported by a grant from IPM (No. 85200038).
‡ Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology, P. O.
Box 16315-1618, Tehran, Iran.

564

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 564-588, August 2009

ELA

http://math.technion.ac.il/iic/ela

565

Generalized Pascal Triangles and Toeplitz Matrices

One can easily verify that (see [2, 6]):
(1.1)

P (∞) = L(∞) · L(∞)t ,

where L(∞) is the infinite unipotent lower triangular matrix


(1.2)

L(∞) = (Li,j )i,j≥0





=





1
1
1
1
1
..
.

1

2
3
4
..
.

1
3 1
6 4
.. ..
. .


1
..
.

..

.













with entries Li,j = ji . We denote by L(n) the finite submatrix of L(∞) with entries
Li,j , 0 ≤ i, j ≤ n − 1.
To introduce our result, we first present some notation and definitions. We recall
that a matrix T (∞) = (ti,j )i,j≥0 is said to be Toeplitz if ti,j = tk,l whenever i − j =
k − l. Let α = (αi )i≥0 and β = (βi )i≥0 be two sequences with α0 = β0 . We shall
denote by Tα,β (∞) = (ti,j )i,j≥0 the Toeplitz matrix with ti,0 = αi and t0,j = βj . We
also denote by Tα,β (n) the submatrix of Tα,β (∞) consisting of the entries in its first
n rows and columns.

We come now back to the (1.1). In fact, one can rewrite it as follows:
P (∞) = L(∞) · I · L(∞)t ,
where matrix I (identity matrix) is a particular case of a Toeplitz matrix.
In [1], Bacher considers determinants of matrices generalizing the Pascal triangle
P (∞). He introduces generalized Pascal triangles as follows. Let α = (αi )i≥0 and
β = (βi )i≥0 be two sequences starting with a common first term α0 = β0 = γ.
Then, the generalized Pascal triangle associated with α and β, is the infinite matrix
Pα,β (∞) = (Pi,j )i,j≥0 with entries Pi,0 = αi , P0,j = βj (i, j ≥ 0) and
Pi,j = Pi−1,j + Pi,j−1 , for i, j ≥ 1.
We denote by Pα,β (n) the finite submatrix of Pα,β (∞) with entries Pi,j , 0 ≤ i, j ≤
n − 1. An explicit formula for entry Pi,j of Pα,β (n) is also given by the following
formula (see [1]):
Pi,j






j
i
i+j
i + j − s

i+j−t
+
=γ
(αs − αs−1 )
+
.
(βt − βt−1 )
j
j
i
s=1
t=1

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 564-588, August 2009

ELA

http://math.technion.ac.il/iic/ela

566

A. R. Moghaddamfar and S. M. H. Pooya

For an arbitrary sequence α = (αi )i≥0 , we define the sequences α
ˆ = (ˆ
αi )i≥0 and
α
ˇ = (ˇ
αi )i≥0 as follows:
α
ˆi =

(1.3)

i


i+k

(−1)

k=0



i
αk
k

i


i
and α
ˇi =
αk .
k
k=0

With these definitions we can now state our main result. Indeed, the purpose
of this article is to obtain a factorization of the generalized Pascal triangle Pα,β (n)
associated with the arbitrary sequences α and β, as a product of a unipotent lower
triangular matrix L(n), a Toeplitz matrix Tα,
ˆ βˆ (n) and a unipotent upper triangular
matrix U (n) (see Theorem 3.1), that is
Pα,β (n) = L(n) · Tα,
ˆ βˆ (n) · U (n).
Similarly, we show that
−1
.
Tα,β (n) = L(n)−1 · Pα,
ˇ βˇ (n) · U (n)

In fact, we obtain a connection between generalized Pascal triangles and Toeplitz
matrices. In view of these factorizations, we can easily see that
det(Pα,β (n)) = det(Tα,
ˆ βˆ (n)).
Finally, we present several applications of Theorem 3.1 to some other determinant
evaluations.
We conclude the introduction with notation and terminology to be used through√
out the article. For convenience, we will let −1 denote the complex number i ∈ C.
By ⌊x⌋ we denote the integer part of x, i.e., the greatest integer that is less than or
equal to x. We also denote by ⌈x⌉ the smallest integer greater than or equal to x.
Given a matrix A, we denote by Ri (A) and Cj (A) the row i and the column j of A,
respectively. We use the notation At for the transpose of A.
In general, an n × n matrix of the following form:


A
C

B
Pα,β (n − k)





resp.



A
C

B
Tα,β (n − k)





where A, B and C are arbitrary matrices of order k × k, k × (n − k) and (n − k) × k,
respectively, is called a quasi-Pascal (resp. quasi-Toeplitz) matrix.
Throughout this article we assume that:

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 564-588, August 2009

ELA

http://math.technion.ac.il/iic/ela

567

Generalized Pascal Triangles and Toeplitz Matrices

F = (Fi )i≥0 = (0, 1, 1, 2, 3, 5, 8, . . . , Fi , . . .)

(Fibonacci numbers),

∗

F = (Fi )i≥1 = (1, 1, 2, 3, 5, 8, . . . , Fi , . . .)

(Fibonacci numbers = 0),

L = (Li )i≥0 = (2, 1, 3, 4, 7, 11, 18, . . . , Li , . . .)

(Lucas numbers),

C = (Ci )i≥0 = (1, 1, 2, 5, 14, 42, 132, . . ., Ci , . . .)

(Catalan numbers),

I = (Ii )i≥0 = (0!, 1!, 2!, 3!, 4!, 5!, 6!, . . . , Ii = i!, . . .)
I ∗ = (Ii )i≥1 = (1!, 2!, 3!, 4!, 5!, 6!, . . . , Ii = i!, . . .)
The rest of this article is organized as follows. In Section 2, we derive some
preparatory results. In Section 3, we prove the main result (Theorem 3.1). Section 4
deals with applications of Theorem 3.1. In Section 5, we present two matrices whose
entries are recursively defined, and we show that the leading principal minor sequence
of these matrices is the sequence F ∗ = (Fi )i≥1 .

2. Preliminary Results. As we mentioned in the Introduction, if α = (αi )i≥0
is an arbitrary sequence, then we define the sequences α
ˆ = (ˆ
αi )i≥0 and α
ˇ = (ˇ
αi )i≥0
as in (1.3). For certain sequences α the associated sequences α
ˆ and α
ˇ seem also to be
of interest since they have appeared elsewhere. In Tables 1 and 2, we have presented
some sequences α and the associated sequences α
ˆ and α
ˇ.
Table 1. Some sequences α and associated sequences α
ˇ.
α
(0, 1, −1, 2, −3, 5, −8, . . .)
(1, 0, 1, −1, 2, −3, 5, −8, . . .)
(2, −1, 3, −4, 7, −11, 18, . . .)
(1, 0, 1, 1, 3, 6, 15, . . .)
(1, 0, 1, 2, 9, 44, 265, . . .)
(1, 1, 3, 11, 53, 309, 2119, . . .)

Reference
A039834 in [8]
A039834 in [8]
A061084 in [8]
A005043 in [8]
A000166 in [8]
A000255 in [8]

α
ˇ
F
F∗
L
C
I
I∗

Table 2. Some sequences α and associated sequences α
ˆ.
α
(0, 1, 3, 8, 21, 55, 144, . . .)
(1, 2, 5, 13, 34, 89, 233, . . .)
(2, 3, 7, 18, 47, 123, 322, . . .)
(1, 2, 5, 15, 51, 188, 731, . . .)
(1, 2, 5, 16, 65, 326, 1957, . . .)
(1, 3, 11, 49, 261, 1631, . . .)

Reference
A001906 in [8]
A001519 in [8]
A005248 in [8]
A007317 in [8]
A000522 in [8]
A001339 in [8]

α
ˆ
F
F∗
L
C
I
I∗

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 564-588, August 2009

ELA

http://math.technion.ac.il/iic/ela

568

A. R. Moghaddamfar and S. M. H. Pooya

As another nice example, consider α = Ri (L(∞)) where L is introduced as in
(1.2). Then, we have
α
ˇ = Ri (P (∞)).
ˇˆ = α.
ˆˇ = α
Lemma 2.1. Let α be an arbitrary sequence. Then, we have α
ˆˇ = (α
ˆˇ i )i≥0 . Then, we have
Proof. Suppose α = (αi )i≥0 and α


i
i+k i
ˆˇ i =
α
ˇk
k=0 (−1)
k α



i
k
k
i+k i
=
k=0 (−1)
s=0 s αs
k



i k
i+k i k
=
k=0
s=0 (−1)
k s αs





i
i
= (−1)i l=0 αl h=l (−1)h hi hl


i−l 
i
i
= (−1)i l=0 αl h=l (−1)h il h−l

 i



h i−l
= (−1)i il=0 αl il
h=l (−1) h−l .

But, if l < i, then we have

i

h i−l
h=l (−1) h−l

ˆˇ i = (−1)i
α

i

l=i






= 0. Therefore, we obtain






i
i
h i−l
αl
= αi ,
(−1)
l
h−l
h=l

ˆˇ = α.
and hence α
The proof of second part is similar to the previous case.
Lemma 2.2. Let i, j be positive integers. Then, we have




 0 if i = j,
i−j

i
k+j
k
(−1)
=
k+j
j
1 if i = j.
k=0

The proof follows from the easy identity








i
k+j
i
i−j
=
.
k+j
j
j
k
The following Lemma is a special case of a general result due to Krattenthaler
(see Theorem 1 in [9]).
Lemma 2.3. Let α = (αi )i≥0 and β = (βj )j≥0 be two geometric sequences with
αi = ρi and βj = σ j . Then, we have
det(Pα,β (n)) = (ρ + σ − ρσ)n−1 .

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 18, pp. 564-588, August 2009

ELA

http://math.technion.ac.il/iic/ela

569

Generalized Pascal Triangles and Toeplitz Matrices

3. Main Result. Now, we are in the position to state and prove the main result
of this article.
Theorem 3.1. Let α = (αi )i≥0 and β = (βi )i≥0 be two sequences starting with
a common first term α0 = β0 = γ. Then, we have
(3.1)

Pα,β (n) = L(n) · Tα,
ˆ βˆ (n) · U (n),

and
−1
Tα,β (n) = L(n)−1 · Pα,
,
ˇ βˇ (n) · U (n)

(3.2)

where L(n) = (Li,j )0≤i,j