Download EVALUATING DETERMINANTS OF CONVOLUTION

c 2007 Institute for Scientific
°
Computing and Information
INTERNATIONAL JOURNAL OF
INFORMATION AND SYSTEMS SCIENCES
Volume 3, Number 4, Pages 569–580
EVALUATING DETERMINANTS OF CONVOLUTION-LIKE
MATRICES VIA GENERATING FUNCTIONS
YONGZHI (PETER) YANG AND MOLLY LEONARD
Abstract. In this paper we extend the results of Getu [2] on evaluating determinants via generating functions to a broader class of matrices, which contain
convolution matrices. As an immediate application of our new findings, we
develop the method of finding the determinant representation of a well-known
sequence. By creating the determinant representations of Chebyshev polynomials of the first kind and Stirling numbers of the second kind, we illustrate
the idea of the method.
Key Words. convolution matrix, generating function, determinant, and sequence.
1. Introduction
In this paper we use generating functions to evaluate the determinants of a
certain class of matrices whose components are convolutions. Our research is an
extension of the method developed by Getu [2]. Because we have greatly generalized
the previous method, we can easily examine a much greater variety of matrices. We
have also gained an increased knowledge in the structure of these matrices which
allows us to easily construct matrices whose determinants, as well as components,
are famous sequences.
2. Convolution Matrices
To review, the convolution of two
Pnsequences {an } and {bn }, for n = 1, 2, 3, . . . ,
is the sequence {cn }, where cn = k=1 ak bn−k+1 . The convolution matrix of two
sequences {an } and {bn } is the matrix whose first column is {an } and whose ith
column (i = 2, 3, . . . ) is the convolution sequence of the (i − 1)th column with {bn }.
We say that the convolution matrix of the sequences {an } and {an } is the convolution matrix of the sequence {an }. There are many well-known integer matrices
which can be written as convolution matrices of some sequences. The rectangular
Pascal matrix, for instance, is the convolution matrix of the sequence {1, 1, 1, 1, . . .}
and the upper triangular Pascal matrix, denoted by PU , is the convolution matrix
of the sequence {1, 0, 0, 0, . . . } and {1, 1, 0, 0, . . . .}.
Received by the editors December 15, 2006 and, in revised form, March 7, 2007.
2000 Mathematics Subject Classification. 15A15, 11B83, 11C20.
This research was supported by the University Scholar Grant of the University of St. Thomas.
569
570
Y. YANG AND M. LEONARD

PU
1
0

0


= 0
0

0

..
.
1
1
0
0
0
0
..
.
1
2
1
0
0
0
..
.
1
3
3
1
0
0
..
.
1 1
4 5
6 10
4 10
1 5
0 1
.. ..
. .

···
· · ·

· · ·

· · ·
.
· · ·

· · ·

..
.
The convolution matrix of two sequences {un } and {vn }, n = 0, 1, 2, 3, . . . , is


u0
u0 v0
u0 v02
···
u1
u0 v1 + u1 v0
2u0 v0 v1 + u1 v02
· · ·


(1) C = u2 u0 v2 + u1 v1 + u2 v0 2u0 v0 v2 + 2u1 v0 v1 + u2 v 2 + u0 v 2 · · · ,
0
1


..
..
..
..
.
.
.
.
and, in [5], its LU decomposition theorem is :
Lemma 2.1. (The Generalized Convolution Decomposition Theorem)
C = SPU v0 for some lower triangular matrix S and the upper triangular Pascal
matrix PU . Morever, S is a convolution matrix of {un } and {0, v1 , v2 , . . . } and
n(n−1)/2
, where Cn is the n × n upper left corner matrix of C.
|Cn | = un0 v1
Furthermore, the convolution matrices of some particular sequences have some
very interesting properties [5] - [7].
3. Case One
The first class of matrices that will be examined consist of the matrices that
have a sequence of numbers juxtaposed on the left side of a convolution matrix in
the first column.


b0 u0
u0 v0
···
à ¯
!
b1 u1
¯
u0 v1 + u1 v0
· · ·

¯

(2)
M= b¯ C
= b2 u2 u0 v2 + u1 v1 + u2 v0 · · · ,
¯


..
..
..
..
.
.
.
.
where b = (b0 , b1 , b2 , . . . bn , . . . )T and C is the convolution matrix of the sequences
{un } = {u0 , u1 , u2 , . . . un , . . . } and {vn } = {v0 , v1 , v2 , . . . vn , . . .}.
To obtain our main theorem, we introduce the following Lemma.
Lemma 3.1.
(3)
det[M] = det[N],
³ ¯
´
¯
where N = b ¯ S
, b = (b0 , b1 , b2 , . . . bn , . . . )T and S is the convolution matrix of the sequences {un } = {u0 , u1 , u2 , . . . un , . . . } and {vn } = {0, v1 , v2 , . . . vn , . . .}.
Proof. By Lemma 2.1, we know that
C = SPU v0 ,
where PU is the upper triangular Pascal matrix and S is a convolution matrix of
{un } and {0, v1 , v2 , . . . }. Thus,
EVALUATING DETERMINANTS OF CONVOLUTION-LIKE MATRICES
³ ¯
´
³ ¯
¯
¯
det[M] = det[ b ¯ C
] = det[ b ¯ SPU v0
#
"Ã ¯
!1 |
0
¯
¯
− −
− 
= det b ¯ S
¯
0 | PU v0
#
" 1 |
0
−  .
= det[N] det − −
0 | PU v 0
(4)
Noting that
" 1
det −
0
|
−
|
571
´
]
#
0
−  =1
PU v0
and using Eq. (4) yields det[M] = det[N]. This completes the proof.
Note: When applying this above relationship to the material we are discussing
in this paper, it is important to keep in mind that C is the convolution matrix of
{un } with {vn } while S is simply the convolution of {un } with {0, v1 , v2 , . . . }. S is
nearly identical to C except that the v0 term is zero.
Theorem 3.1. Consider an infinite dimensional matrix of the following form


b0 u0
u0 v0
···
à ¯
!
b1 u1
¯
u0 v1 + u1 v0
· · ·

¯

(5)
M= b¯ C
= b2 u2 u0 v2 + u1 v1 + u2 v0 · · ·  ,
¯


..
..
..
..
.
.
.
.
where b = (b0 , b1 , b2 , . . . bn , . . . )T and C is the convolution matrix of the sequences:
, . . .}. Let
2 , . . . un , . . . } and {v
Pn∞} = {v0 , v1 , v2 , . . . vnP
P∞{un } = {u0 , u1 , uP
∞
∞
A(x)= n=1 an xn−1 , B(x)= n=0 bn xn , U(x)= n=0 un xn , and V(x)= n=0 vn xn
be the generating functions for the sequences {an }, {bn }, {un }, and {vn }, respecn n(n−1)/2
tively. If A[V (x)] = B(x)
an = (−1)n−1 Dn , where Dn is the
U (x) , then u0 v1
determinant of the n × n upper left corner matrix Mn of the original matrix M.
Proof. Consider the following system of equations

 a  b 
1
0
b1 
a2 






   

 a3  = b2 
C
(6)
.

   

 a4  b3 

..
..
.
.
If {U (x)} and {V (x)} are generating functions for {un } and {vn }, respectively,
then it is well known that the column generating functions of the convolution matrix
C can be written this way

¡
1
x x2
x3
. .

¢
. 




 ¡
C 
 = U (x)

U (x)V (x) U (x)V 2 (x)
¢
··· .
572
Y. YANG AND M. LEONARD
Noting that B(x) is the generating function for {bn } and multiplying the vector
¡
¢T
1 x x2 x3 . . . to Eq. (6) results in

 a 
1

a


2

¡
¢
a3 
2
3


1 x x x . . . 
C





a4 
..
.
 
(7)
a1
a2 

¡
¢
 
= U (x) U (x)V (x) U (x)V 2 (x) U (x)V 3 (x) · · · a3 
a4 
 
..
.
= B(x).
Thus, we have
U (x)(a1 + a2 V (x) + a3 V 2 (x) + a4 V 3 (x) . . . ) = B(x),
which yields
(8)
U (x)A[V (x)] = B(x).
Therefore, it is clear that A[V (x)] does indeed equal
B(x)
U (x) .
To derive the second part of Theorem 3.1, we consider the subsystem of equations
in Eq. (6)

 a   b 
1
0
  b1 
a


2
 



 b2 
a3 


C
(9)
=
,



n

 .   . 

 .   . 

.
.
an
bn−1
where Cn is the n × n upper left corner matrix of C. Let’s solve the system of
equations in Eq. (9) for an using Cramer’s Rule

|

b0
.. 
. 

b0

 ..
n−1
det  C̄n |
(−1)
det  .
| bn−1
bn−1
an =
=
|Cn |
|Cn |
(−1)n−1 Dn
(−1)n−1 det[Mn ]
=
,
=
|Cn |
|Cn |
|
| C̄n
|



where C̄n is the n × (n − 1) matrix obtained by eliminating the nth column of
Cn and Mn is the n × n upper left corner matrix of the matrix M. According to
n(n−1)/2
n(n−1)/2
Lemma 2.1, |Cn | = un0 v1
, so it is clear that un0 v1
an = (−1)n−1 Dn .
This completes the proof.
The Proposition 1 in [2] is an immediate consequence of Theorem 3.1.
EVALUATING DETERMINANTS OF CONVOLUTION-LIKE MATRICES
573
Corollary 3.1. Consider the following infinite matrix with 1’s in the super diagonal.


b0 1 0 0 · · ·
b1 c1 1 0 · · · 




D = b2 c2 c1 1 · · ·  .
b3 c3 c2 c1 · · · 


..
..
.. . .
..
.
.
.
.
.
P∞
P∞
Let B(x)= n=0 bn xn and C(x) = n=1 cn xn be the generating functions for the
B(x)
sequences {b0 , b1 , b2 , . . .} and {c1 , c2 , c3 , . . .}, respectively. If A(x) = 1+C(x)
=
P∞
n
n−1
a
x
,
then
a
=
(−1)
D
and
1
+
xA(−x)
is
the
generating
function
of
n
n
n=0 n+1
Dn , which is the determinant of the n × n upper left corner matrix of the matrix
D.
Proof. Clearly, when u0 , u1 , u2 , u3 , . . . and v0 , v1 , v2 , v3 , . . . in Theorem 3.1 are
1, c1 , c2 , c3 , . . . and 0, 1, 0, 0, . . . , respectively, M in Theorem 3.1 equals D. Thus,
the corollary follows readily from Theorem 3.1.
Example 3.1 Consider the following matrix

(10)
1
4

9


R = 16
25

36

..
.
1
1
0
0
0
0
..
.
1
2
1
0
0
0
..
.
1
3
3
1
0
0
..
.
1
4
6
4
1
0
..
.
1
5
10
10
5
1
..
.
1
6
15
20
15
6
..
.

···
· · ·

· · ·

· · ·
.
· · ·

· · ·

..
.
We wish to determine the upper left corner determinants. We can set {bn } equal
to the first column of our matrix R. {bn } = {1, 4, 9, 16, 25, 36, . . . } = {12 , 22 , 32 , 42 ,
52 , 62 , . . . }. We notice that these numbers are simply the square positive integers
and can be described by the generating function B(x) = x(x+1)
(1−x)3 . The remaining
part of the matrix is the truncated Pascal’s upper triangular matrix which is a convolution of the sequence {un } = {1, 1, 0, 0, 0, . . . } and {vn } = {1, 1, 0, 0, 0, 0, . . . }.
So that means U (x) = 1 + x and V (x) = 1 + x. Plugging these generating functions
into Theorem 3.1 yields
x
x(x + 1)
=
3
(1 − x) (x + 1)
(1 − x)3
We are now left with A(x+1) but we wish to know {an } which is modeled by the
generating function A(x). In order to resolve this problem we can safely let v0 = 0
without changing the value of the determinant (this is proven in Lemma 3.1). Now
that v0 = 0 , {vn } = {0, 1, 0, 0, 0, . . . } and V (x) simply becomes x. Thus, we have
the new equation:
A(x + 1) =
x
= x + 3x2 + 6x3 + 10x4 + 15x5 . . . ,
(1 − x)3
which is the generating function for the triangular number sequence, i.e., {an } =
n(n−1)/2
{Tn }, where Tn is the nth triangular number. From Theorem 3.1: u0 v1
an =
n−1
(−1)
Dn . In this example both u0 and v1 equal one so that
A[V (x)] = A[x] =
Dn = det[Rn ] = (−1)n−1 Tn .
574
Y. YANG AND M. LEONARD
It is also very interesting to note that the triangular numbers are the part of the
third row of our convolution matrix, as well as the second super diagonal. So
Dn = det[Rn ] = (−1)n−1 R3,n+2 = (−1)n−1 Rn,n+2
and the determinants of the upper left corner matrices of R can be determined by
simply looking at the third row of the original matrix.
All the matrices examined in [1] can be explained using Theorem 3.1. This article
dealt with developing matrices whose determinants were Fibonacci numbers. Let’s
examine their first example and consider the matrix


1 i 0 0 ...
 i 1 i 0 . . .


0 i 1 i . . . 
.
(11)
Φ=


0 0 i 1 . . . 


.. .. .. . .
..
.
.
. . .
Corollary 3.2.
det[Φn ] = Fn+1 , for n ≥ 1,
where Fn is the nth term of Fibonacci number sequence and Φn is the n × n upper
left corner matrix of Φ .
Proof. From the structure of the matrix Φ, we see that
U (x) = i + x + ix2 ;
B(x) = 1 + ix;
V (x) = x.
Using Theorem 3.1 yields
(12)
A[x] =
B(x)
1 + ix
1
1 + (ix)
=
=
.
U (x)
i + x + ix2
i 1 − (ix) − (ix)2
n(n−1)/2
By Theorem 3.1, we know det[Φn ] = Dn = (−1)n−1 un0 v1
Therefore,
(13)
∞
X
n−1
det[Φn ]x
=
n=1
∞
X
n=1
n−1
Dn x
=
∞
X
an = (−1)n−1 in an .
ian (−ix)n−1 = iA[−ix].
n=1
Combining Eqs. (12)-(13) leads
(14)
∞
X
det[Φn ]xn−1 =
n=1
∞
X
n=1
ian (−ix)n−1 = iA[−ix] =
1+x
,
1 − x − x2
which equals the generating function for truncated Fibonacci number sequence
{F2 , F3 , F4 , F5 , . . .} = {1, 2, 3, 5, . . .} . This completes the proof.
Not only are we able to determine the determinants of any matrix of the form
specified in Theorem 3.1, but we can easily construct a matrix whose determinants
are a famous sequence and have some generating function, K(x). K(x) represents
the desired generating function for the determinants. We will illustrate the idea in
the following example.
Example 3.2 We use the generating function for Fibonacci number sequence and
set
K(x) =
x
= x + x2 + 2x3 + 3x4 + 5x5 + . . . .
1 − x − x2
EVALUATING DETERMINANTS OF CONVOLUTION-LIKE MATRICES
575
Thus, by Theorem 3.1, we have
B(x)
= K(x).
U (x)
Let’s next choose U(x) and V(x). We will make U (x) be the generating function
for Catalan number sequence and V (x) can be a simple polynomial.
A[V (x)] =
U (x) =
1−
√
1 − 4x
= 1 + x + 2x2 + 5x3 + 14x4 + . . . and V (x) = 2 + x.
2x
Keeping in mind the relationship
B(x)
U (x)
= K(x) , B(x) must equal K(x) times U (x).
√
√
x(1 − 1 − 4x)
( 1 − 4x − 1)
B(x) = K(x)U (x) =
=
(1 − x − x2 )2x
2(x2 + x − 1)
= x + 2x2 + 5x3 + 12x4 + 31x5 + . . . ,
which happens to be the generating function for the convolution of Catalan and
Fibonacci numbers. The matrix created in this example looks like this:

0 1 2
1 1 3

2 2 5

 5 5 12

12 14 33

31 42 98

..
..
..
.
.
.
4
8
13
29
78
229
..
.
8
20
34
71
185
536
..
.

16
32 · · ·
48
112 · · ·

88
224 · · ·

176 440 · · ·

441 1058 · · ·

1257 2955 · · ·

..
..
..
.
.
.
and the upper left corner determinants can indeed be modeled by the generating
function for Fibonacci number sequence (which is A(x), and because u0 and v1 are
both one, an = (−1)n−1 Dn ). Therefore,
¯
¯
¯
¯ ¯
4 ¯¯
¯
¯ ¯0 1 2 ¯ ¯0 1 2
¯ ¯ ¯0 1¯ ¯
¯ ¯
8 ¯¯
¯ , ¯1 1 3 ¯ , ¯1 1 3
D1 , D2 , D3 , D4 , · · · = ¯0¯ , ¯¯
¯ ¯2 2 5 13¯ , . . .
1 1¯ ¯¯
¯
2 2 5¯ ¯¯
5 5 12 29¯
= 0, −1, 1, −2, 3, −5, · · ·
= F0 , −F1 , F2 , −F3 , F4 , −F5 , · · · .
Note: There are an infinite number of ways to choose the generating functions
and end up with the determinants being the desired sequence. The only restriction
is that the relationship among all the generating functions described in Theorem
3.1 remains true. This example could have just as easily been created by choosing
B(x) first and then determining the corresponding convolution matrix.
As seen in the previous example, C was the convolution matrix of the Catalan
number sequence with the simple polynomial 2+x, A(x) was the generating function
of Fibonacci number sequence, and B(x) was therefore the generating function of
the convolution between the Catalan numbers and the Fibonacci numbers. This is
a very interesting relationship that is present as long as the generating function for
the sequence used to make the convolution matrix starts with one, and is convoluted
with a sequence whose generating function has the form v0 + x, where v0 is any
constant value.
576
Y. YANG AND M. LEONARD
Consider a special case of the matrices examined in Theorem 3.1. The convolution matrix C is composed of a convolution of any sequence {un } with the
sequence {v0 , 1, 0, 0, . . . }. The {bn } sequence in the first column of the original
matrix is a convolution of the sequence {un } with any sequence {(−1)n zn }. We
have the following result.
Theorem 3.2. Let the special matrix W be
÷
¸!
¸ ¯¯ ·
Convolution M atrix
Convolution sequence of ¯
.
W=
¯
of {un } and {v0 , 1, 0, 0 . . . }
{(−1)n zn } and {un }
¯
If u0 = 1, then det[Wn ] = zn−1 , for n = 1, 2, 3 . . . , where Wn is the n × n upper
left corner matrix of W .
Proof. By Theorem 3.1 we have A[V (x)] = B(x)
U (x) . Noting B(x) = Z(−x)U (x) and
V (x) = x (by Lemma 3.1, we can choose V (x) = x instead of V (x) = v0 + x) yields
A(x) =
Z(−x)U (x)
= Z(−x).
U (x)
P∞
P∞
P∞
n n
n−1
det[Wn ]xn−1 =
Now, n=1 an xn−1 =
n=0 zn (−1) x . Comn=1 (−1)
l−1
paring the coefficients of x
leads to det[Wl ] = zl−1 , for l = 1, 2, 3, . . . .
Remark: Any sequence {zn } can be represented in terms of a sequence of determinants.
The relationship found in Theorem 3.2 can be used in a very interesting application of Chebyshev polynomials. The Chebyshev polynomials of the first kind
{Hn (x)} are listed as follows:
H0 (x) = 1
H1 (x) = x
H2 (x) = 2x2 − 1
H3 (x) = 4x3 − 3x
H4 (x) = 8x4 − 8x2 + 1
H5 (x) = 16x5 − 20x3 + 5x
..
.
P∞
1−xt
n
The generating function for {Hn (x)} is 1−2xt+t
2 =
n=0 Hn (x)t and the gener1+xt
1+xt
n
ating function for {(−1) Hn (x)} is 1+2xt+t2 . Choose 1+2xt+t2 to be the generating
function of {(−1)n zn } in Theorem 3.2. Thus, letting {un } ’s generating function
be 1 + 2xt + t2 and {v0 , 1, 0, 0, . . . }’s generating function be t and using Theorem
3.2 yields
EVALUATING DETERMINANTS OF CONVOLUTION-LIKE MATRICES
Corollary 3.3.
¯
¯1 1
¯
¯
¯x 2x
¯
¯
¯0 1
¯
¯
Hn (x) = ¯ 0 0
¯
¯
¯0 0
¯
¯. .
¯ ..
..
¯
¯0 0
0
0
1
0
2x
1
1
2x
0
..
1
..
.
0
0
.
...
..
.
..
.
..
.
..
.
..
.
...
0
0
0
0
1
..
1
.
¯
0¯
¯
¯
0 ¯¯
¯
0 ¯¯
¯
,
0 ¯¯
¯
0 ¯¯
¯
1 ¯¯
2x¯(n+1)×(n+1)
577
n = 0, 1, 2, . . . .
This is a generalization of the well-known determinant equation that classifies
the Chebyshev polynomials of the first kind in [3]:
¯
¯x
¯
¯
¯1
¯
¯
¯0
¯
¯
Hn (x) = ¯ 0
¯
¯
¯0
¯
¯.
¯ ..
¯
¯0
1
0
0
2x
1
0
1
2x
1
0
1
2x
0
..
.
0
..
.
0
0
1
..
.
0
...
..
.
..
.
..
.
..
.
..
.
...
0
0
0
0
1
..
.
1
¯
0¯
¯
¯
0 ¯¯
¯
0 ¯¯
¯
,
0 ¯¯
¯
0 ¯¯
¯
1 ¯¯
2x¯n×n
n = 1, 2, 3, . . . .
4. Case Two
The second class of matrices examined in this paper have all the same properties
as the first class of matrices, except that the super diagonal matrix (1, 2, 3, 4, . . . )
is subtracted from the matrix.
Consider the following infinite dimensional matrix:
à ¯
!
¯
¯
J= b¯ E
,
¯
where b is the sequence {b0 , b1 , b2 , . . . bn , . . .} and E is G−Q, G being a convolution
matrix of the sequence {un } = {u0 , u1 , u2 , . . . } and {vn } = {0, v1 , v2 , . . . } and Q
being the diagonal matrix diag(1, 2, 3, . . .).


b0 u0 − 1
0
···
b1
u1
u0 v1 − 2
· · ·


J = b2
.
u
u
v
+
u
v
·
·
·
2
0
2
1
1


..
..
..
..
.
.
.
.
P∞
P∞
P∞
Theorem 4.1. Let B(x)= n=0 bn xn , U(x)= n=0 un xn , and V(x)= n=1 vn xn
be the generating functions for the sequences {bn }, {u
Pn∞}, and {vn }, respectively.
And let F(x) = f0 + f1 x + f2 x2 + . . . + fn xn + . . . = n=0 fn xn . If
µ
¶
U (x)
0
F [V (x)] − f0 = B(x),
F (x) −
V (x)
578
Y. YANG AND M. LEONARD
n
Qn
|Jn |
then fn = (−1)
, where |En | = i=1 (u0 v1i−1 − i ) and |Jn | is the determinant
|En |
of the n × n upper left corner matrix Jn of J.
Proof. Consider the following system of equations
(15)

u0 − 1
 u1

 u2

..
.
0

  
−f1
b0
···
−f2  b1 
· · ·

  

 =  .
· · ·
 −f3  b2 
..
..
..
.
.
.
0
0
u0 v1 − 2
u0 v2 + u1 v1
..
.
u0 v12 − 3
..
.
Or more simply
¡ ¢¡ ¢ ¡ ¢
E f = b ,
(16)
where f = (−f1¡, −f2 , −f3 , . . . , −fn ,¢. . .)T .
Multiplying 1 x x2 x3 . . . to the system (16) from the left yields
¡
1 x x2
Distributing results in
(17)
h¡
U (x) U (x)V (x)
x3
. .
¢¡
¢
. G − Q f = B(x).
U (x)V 2 (x) .
¢ ¡
. . − 1
2x
3x2
¢i
. . . f
= B(x).
Eq. (17) can be written as
(18)
−U (x)
∞
X
fn V n−1 (x) +
n=1
∞
X
fn nxn−1 = B(x).
n=1
Thus, Eq. (18) is equivalent to
·
¸
U (x)
F (x) −
F [V (x)] − f0 = B(x).
V (x)
0
n
|Jn |
Using Cramer’s Rule, fn = (−1)
can be derived in the same fashion as case
|En |
one matrices. This completes the proof.
Theorem 4.1 covers Proposition 2 examined by Getu [2]. Getu examined matrices
of the following form:
(19)

c1
c2

c3

Ω = c4

c5

..
.
−1
c1
c2
c3
c4
..
.
0
−2
c1
c2
c3
..
.
0
0
−3
c1
c2
..
.

···
· · ·

· · ·

· · ·

· · ·

..
.
and reached the following conclusion.
P∞
P∞
Corollary 4.1. Let C̃(x) = n=1 cn xn−1 and let F (x) = 1 + n=1 fn xn be the
generating function for some numbers such that F 0 (x) = C̃(x)F (x). Then |Ωn | =
fn , where |Ωn | is the determinant of the n × n upper left corner matrix Ωn of Ω.
EVALUATING DETERMINANTS OF CONVOLUTION-LIKE MATRICES
579
Proof. Let the sequences {b0 , b1 , b2 , b3 , . . .}, {u0 , u1 , u2 , u3 , . . .}, and {0, v1 , v2 , v3 , . . .}
in Theorem 4.1 be the sequences {c1 , c2 , c3 , . . .}, {0, c1 , c2 , c3 , . . .}, and {0, 1, 0, 0, . . .},
respectively. Then, B(x), U (x), and V (x) in Theorem 4.1 become
B(x) =
∞
X
bn xn =
n=0
∞
X
cn+1 xn =
n=0
∞
X
cj xj−1 = C̃(x),
j=1
V (x) = x,
and
U (x) =
∞
X
un xn =
n=0
∞
X
cn xn = xC̃(x).
n=1
Making the above substitutions to Theorem 4.2 and letting f0 = 1 leaves us with
xC̃(x)
[F (x) − 1] = C̃(x).
x
Therefore, F 0 (x) = C̃(x)F (x). This completes the proof.
The following example is a very interesting application, which provides a new
way to generate Stirling numbers of the second kind.
Example 4.1 As we know [4], Stirling numbers {Sn,k } of the second kind have the
following exponential generating functions
F 0 (x) −
∞
X
Sn,k xn
(ex − 1)k
=
n!
k!
n=0
Using Theorem 4.1 and letting
U (x) = xk;
V (x) = x;
and
B(x) =
(ex − 1)k−1
,
(k − 1)!
where B(x) actually happens to be the exponential generating function for Sn,k−1 .
Here is the matrix we created

(20)
S0,(k−1) /0!
S1,(k−1) /1!

S2,(k−1) /2!

∆ = S3,(k−1) /3!

S4,(k−1) /4!

..
.
−1
k
0
0
0
..
.
0
−2
k
0
0
..
.
0
0
−3
k
0
..
.
0
0
0
−4
k
..
.

...
. . .

. . .

. . .

. . .

..
.
and the determinants of its upper left corner submatrices are S1,k , S2,k , S3,k , . . . ,
i.e., |∆n | = Sn,k .
How does this work? Using Theorem 4.1
"
#
U (x)
0
F (x) −
F [V (x)] − f0 = B(x)
V (x)
and plugging in the generating functions as we have defined them in this example
gives us:
"
#
xk
(ex − 1)k−1
0
F (x) −
F (x) − f0 =
x
(k − 1)!
580
Y. YANG AND M. LEONARD
(ex − 1)k−1
(k − 1)!
Assume that F (0) = f0 = 0. Then, the linear first order differential equation can
be simplified to:
F 0 (x) − kF (x) + kf0 =
(21)
F 0 (x) − kF (x) =
(ex − 1)k−1
.
(k − 1)!
The solution to Eq. (21) is the exponential generating function for Stirling numbers
of the second kind.
∞
X
Sn,k xn
(ex − 1)k
=
.
F (x) =
(k)!
n!
n=0
Using Theorem 4.1 and noting |En | = (−1)n n! yields
Sn,k
|∆n | = (−1)n
(−1)n n! = Sn,k .
n!
5. Conclusion
These results take a previous method of evaluating determinants and extend that
method developed in [2] to include convolution matrices of any nature. This generalization is very useful because convolutions are found in numerous applications
of mathematics and engineering.
References
[1] Nathan Cahill, John D’Errico, Darren Narayan, and Jack Narayan, Fibonacci Determinants,
The College Mathematics Journal, 33 (2004), 221-225.
[2] Seyoum Getu, ‘Evaluating Determinants via Generating Functions, Mathematics Magazine,
64 (1991), 45 - 53.
[3] A. Rabenstein, Introduction to Differential Equations, Academic Press, New York, 1972.
[4] K. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, New York,
2000.
[5] Yongzhi Yang, Generating Functions of Convolution Matrices. In Applications of Fibonacci
Numbers (Ed. Fredric T. Howard), 9 (2004), Kluwer Academic Publishers, 289-295.
[6] Yongzhi Yang and Johann Leida, Pascal Decompositions of Arithmetic and Convolution Arrays
in Matrices, The Fibonacci Quarterly, 40 (2002), 136-295.
[7] Yongzhi Yang and Johann Leida, Pascal Decompositions of Geometric Arrays in Matrices, The
Fibonacci Quarterly, 42 (2004), 205-215.
Department of Mathematics, University of St. Thomas, 2115 Summit Ave., St. Paul, MN
55105-1079, USA
E-mail: [email protected]
E-mail: [email protected]