/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 8, No. 1, 2015, 135-151

ISSN 1307-5543 – www.ejpam.com

Observations on Some Special Matrices and Polynomials

Moawwad El-Mikkawy!, Faiz Atlan

Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Abstract. In the current paper we focus on the study of three special matrices and two symmetric
polynomials. As a consequence, a recurrence relation satisfied by the entries of the n " n inverse
matrix, Qn of the n" n symmetric Pascal matrix, Pn is obtained. Moreover, a new proof for El-Mikkawy
conjecture [14] is investigated. Finally, some identities are discovered.

2010 Mathematics Subject Classifications: 15A23, 15Bxx, 11B73, 05E05, 68W30, 65F05

Key Words and Phrases: Pascal matrix, Vandermonde matrix, Stirling matrix, Permutation matrix,
Divided difference, Symmetric polynomials, MAPLE

1. Introduction and Basic Definitions

Matrices play an important role in all branches of science, engineering, social science and

management. Matrices are used for data classification by which many problems are solved us-

ing computers. There are many special types of matrices such as the Pascal [1], Vandermonde

[10], Stirling [12], and others. These matrices are of specific importance in many scientific

and engineering applications. For example the Pascal matrix, which has been known since

1303, but has been studied carefully only recently [1], appears in combinatorics, image pro-

cessing, signal processing, numerical analysis, probability and surface reconstruction. Much

researches has been devoted to deal with such matrices and relations between them (see for

instance, [3–5, 7, 9, 19, 21–25, 29–37]). The main objectives of the current paper is to intro-

duce a recurrence relation for the inverse of the symmetric Pascal matrix, Pn of order n, some

new identities and to give another proof for El-Mikkawy conjecture [14]. The paper is orga-

nized as follows: The main results are given in the next section. In Section 3, we present a new

proof for El-Mikkawwy conjecture [14]. Moreover, new identities are obtained. Throughout

this paper, !i j is the Kronecker delta which is equal to 1 or 0 according as i = j or not. Also
!

n

r

"

denotes the binomial coefficient and X denotes the set {x1, x2, . . . , xn}.

!Corresponding author.

Email addresses: m#elmikkawy@yahoo.com, (M. El-Mikkawy), faizatlan11@yahoo.com (F. Atlan)

http://www.ejpam.com 135 c$ 2015 EJPAM All rights reserved.



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 136

Definition 1. [6]. For integer numbers n and k with n% k % 0, the Stirling numbers of the first

kind, s(n, k) and of the second kind, S(n, k) are defined respectively by:

(x)n = [x # n+ 1]n =

n
#

k=0

s(n, k)xk, (1)

and

xn =

n
#

k=0

S(n, k)(x)k, (2)

where the falling factorial of x, (x)n and the rising factorial of x, [x]n are given respectively by:

(x)n =

$

1 if n= 0,

x(x # 1)(x # 2) . . . (x # n+ 1) if n% 1,
(3)

and

[x]n =

$

1 if n= 0,

x(x + 1)(x + 2) . . . (x + n# 1) if n% 1
. (4)

It is well known that for integers n, k % 0, the s(n, k) and S(n, k) satisfy the following

Pascal-type recurrence relations:

s(n, k) = s(n# 1, k# 1)# (n# 1)s(n# 1, k), (5)

and

S(n, k) = S(n# 1, k# 1) + kS(n# 1, k), (6)

subject to:

s(k, k) = S(k, k) = 1,1& k & n (7)

and

s(n, k) = S(n, k) = !nk, if k = 0 or n= 0. (8)

Replace x by #x in (1) gives

[x]n =

n
#

k=0

c(n, k)xk, (9)

where c(n, k) = (#1)n#ks(n, k) is called the unsigned or signless Stirling number of the first

kind. It can be shown that the rising factorial, [x]n and the falling factorial, (x)n are also

related by:

[x]n =

n
#

k=0

H(n, k)(x)k, (10)

where

H(n, k) =
n!

k!

!

n# 1

k# 1

"

. (11)



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 137

The coefficients, H(n, k) in (10) are called the Lah numbers. These numbers satisfy the fol-

lowing Pascal-type recurrence relation:

H(n, k) = H(n# 1, k# 1) + (n+ k# 1)H(n# 1, k). (12)

The Lah numbers also satisfy:

H(i, j) =

n
#

k=1

(#1)i+ks(i, k)S(k, j), 1& i, j & n. (13)

Definition 2 ([6]). The Stirling matrix of the first kind, sn and of the second kind, Sn are defined

respectively by:

sn =

$

s(i, j) for i % j,

0 otherwise
, (14)

and

Sn =

$

S(i, j) for i % j,

0 otherwise
. (15)

For example

s5 =

%

&

&

&

&

'

1 0 0 0 0

#1 1 0 0 0

2 #3 1 0 0

#6 11 #6 1 0

24 #50 35 #10 1

(

)

)

)

)

*

and S5 =

%

&

&

&

&

'

1 0 0 0 0

1 1 0 0 0

1 3 1 0 0

1 7 6 1 0

1 15 25 10 1

(

)

)

)

)

*

.

It is known that snSn = Snsn = In, where In is the n" n identity matrix. In other words, the

matrices, sn and Sn are inverse to each other. If fi(x) = x i , 1& i & n, then

S(i, k) = fi[0,1,2, . . . , k] =
!k fi(0)

k! , where fi[0,1,2, . . . , k] is the k-th divided difference of the

function fi(x) = x i which lies on the top of each column and ! is the forward difference

operator. For example, for f5(x) = x5, we have:

Table 1: Divided Di!erences and Stirling Numbers of the Second Kind

x f5(x) First DD Second DD Third DD Fourth DD Fifth DD

0 0 1=S(5,1) 15=S(5,2) 25=S(5,3) 10=S(5,4) 1=S(5,5)

1 1 31 90 65 15

2 32 211 285 125

3 243 781 660

4 1024 2101

5 3125

Thus,

f5(x) = x5 = (x)1 + 15(x)2 + 25(x)3 + 10(x)4 + (x)5



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 138

and

! f5(0) = 1,!2 f5(0) = 30,!3 f5(0) = 150,!4 f5(0) = 240,

and

!5 f5(0) = 120.

Definition 3 ([16]). The elementary symmetric polynomial "(n)r and the complete symmetric

polynomial #(n)r in x1, x2, . . . , xn are defined respectively by:

"(n)r (x) :=

+

,
,

-

,
,

.

0 if r > n or n< 0 or r < 0,

1 if r = 0,
/

1&i1<i2<...<ir&n

xi1
xi2

. . . xir
if 1& r & n.

(16)

and

#(n)r (X) :=

+

,
,

-

,
,

.

0 if r < 0 or n< 0 or (n= 0 and r > 0),

1 if r = 0,
/

1&i1&i2&...&ir&n

xi1
xi2

. . . xir
if r % 1.

(17)

It should be noticed that each"(n)r has

!

n

r

"

terms and each #(n)r has

!

n+ r # 1

r

"

terms.

Moreover these polynomials can be expressed as:

"(n)r (x) :=
#

k1+k2+...+kn=r

k1,k2,...,kn'{0,1}

x
k1

1 x
k2

2 . . . xkn
n , 0& r & n. (18)

and

#(n)r (X) :=
#

d1+d2+...+dn=r

d1,d2,...,dn'{0,1,...,r}

x
d1

1 x
d2

2 . . . xdn
n , r % 0. (19)

The falling factorial (x)n in (3) can be written in the form:

(x)n =

n
#

k=0

(#1)n#k"
(n)
n#k
(0,1, . . . , n# 1)xk. (20)

Comparing the coefficients of xk in (1) and (20), yields

c(n, k) = "
(n)
n#k
(0,1, . . . , n# 1) = "

(n#1)
n#k

(1,2, . . . , n# 1). (21)

It can also be shown that

S(n, k) = #
(k)
n#k
(1,2, . . . , k). (22)

Also,

S(n, k) =
1

k!

k
#

j=0

(#1)k# j

!

k

j

"

jn. (23)



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 139

For any x j ' X, we have

"
(n)
i (x1, x2, . . . , xn) ="

(n#1)
i (x1, x2, . . . , x j#1, x j+1, . . . , xn)

+ x j"
(n#1)
i#1 (x1, x2, . . . , x j#1, x j+1, . . . , xn). (24)

Partial differentiation for both sides of (24) with respect to x j , j ' {1,2, . . . , n} gives

$

$ x j

"
(n)
i = "

(n)
i, j = "

(n#1)
i#1 (x1, x2, . . . , x j#1, x j+1, . . . , xn). (25)

Therefore by using (16), we obtain

"
(n)
i j (X) =

+

,
,
,

-

,
,
,

.

0 if i > n or i < 0 or n< 0,

1 if i = 1,
/

1&r1<r2<...<ri#1&n

r1,r2...,ri#1 (= j

xr1
xr2

. . . xri#1
if 2& i & n.

(26)

Definition 4 ([17]). The Vandermonde matrix V of order n is a matrix of the form:

V =
0

x i#1
j

1n

i, j=1
. (27)

The Vandermonde determinant formula is well known, see for instance [26], in many text

books and articles. It is given by:

det(V ) =
2

2

2x i#1
j

2

2

2

n

i, j=1
=
3

1& j<i&n

(xi # x j). (28)

From (28), we see that if the xi are distinct, then this determinant, det(V ) is nonzero and hence

V is invertible. The explicit form of the inverse matrix, V#1 = (%i j)
n
i, j=1 of the Vandermonde

matrix V is given by [11]:

%i j =(#1)n# j
"
(n)
n# j+1,i(x1, x2, . . . , xn)

Fi

(29)

= (#1)n# j
"
(n#1)
n# j (x1, x2, . . . , xi#1, xi+1, . . . , xn)

Fi

, 1& i, j & n, (30)

where

Fi =

n
3

r=1
r (=i

(xi # xr), i = 1,2, . . . , n. (31)

Consequently, the cost of the solution of the Vandermonde linear system

V [u1u2 . . . un]
T = [d1d2 . . . dn]

T (32)



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 140

is O(n2). Setting xk = k, 1& k & n, in (30), yields:

%i j =
(#1)i+ j

(n# 1)!

!

n# 1

i # 1

"

"
(n#1)
n# j (1,2, . . . , i # 1, i + 1, . . . , n), 1& i, j & n. (33)

The Vandermonde matrix, V in (27) satisfies:

V = LU , (34)

where L = (Li j)
n
i, j=1 is an n" n lower triangular matrix given by:

Li j =

4

#
( j)
i# j(x1, x2, . . . , x j) for i % j

0 for i < j
, 1& i, j & n (35)

and U = (Ui j)
n
i, j=1 is an n" n upper triangular matrix given by:

Ui j =

+

-

.

0 for i > j,
i#1
5

r=1

(x j # xr) for i & j
, 1& i, j & n (36)

The inverse matrices L#1 and U#1 of the matrices L and U in (35) and (36) are given respec-

tively by:

L#1 =

6

(#1)i# j"
(i#1)
i# j (x1, x2, . . . , xi#1)

7n

i, j=1
(37)

and

U#1 =

6

1
j
5

r=1
r (=i

(xi # xr)

7n

i, j=1
. (38)

In particular, if xk = k, 1& k & n, then we have [6]:

V = Sn L̃T ,

where Sn is the Stirling matrix of the second kind and L̃ = ( L̃i j)
n
i, j=1 is an n"n lower triangular

matrix given by:

L̃i j =

$

[i # 1] j#1 for i % j

0 for i < j
, 1& i, j & n (39)

The inverse of the matrix L̃ is given by:

L̃#1 =

6

(#1)i# j 1

(i # 1)!

!

i # 1

j # 1

"
7n

i, j=1
.

Definition 5 ([18]). An n" n matrix A is called totally positive if all its minors of all sizes are

positive.



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 141

Definition 6 ([8]). The symmetric matrix A= (ai j)
n
i, j=1 is called positive definite if and only if

xT Ax> 0, for all x ' !n, x (= 0.

Definition 7. The n" n permutation matrix, J is defined by:

J =
8

en, en#1, . . . , e1

9

, (40)

where ei =
8

!i1,!i2, . . . ,!in

9T
, and !i j is the kronecker delta.

The permutation matrix J of order n enjoys the following properties:

• J = J T = J#1.

• Jk =

$

In k even

J k odd
, where In is the identity matrix of order n.

• det(J) = (#1)
n(n#1)

2 =

$

1 if n) 0 or 1 mod(4)

#1 if n) 2 or 3 mod(4).
.

Lemma 1. For nonnegative integer numbers n and k, we have

n
#

r=1

!

r

k

"

=

!

n+ 1

k+ 1

"

#!k0. (41)

Proof. By using the following Pascal’s rule:

!

r

k

"

=

!

r + 1

k+ 1

"

#

!

r

k+ 1

"

,

we get:

n
#

r=1

!

r

k

"

=

n
#

r=1

6
!

r + 1

k+ 1

"

#

!

r

k+ 1

"
7

=

!

n+ 1

k+ 1

"

#

!

1

k+ 1

"

=

!

n+ 1

k+ 1

"

#!k0,

having used the telescoping sum.

Definition 8 ([20]). A real n" n matrix A= (ai j)
n
i, j=1 is called row stochastic matrix if

(i) ai j % 0, for 1& i, j & n

(ii)
/n

j=1 ai j = 1, for 1& i & n



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 142

Note that (ii) is equivalent to AE = E, where E = [1,1, . . . , 1]T .

Definition 9 ([15]). The symmetric Pascal matrix Pn = (Pi j)
n
i, j=1 of order n is a matrix of integers

defined by:

Pi j =

!

i + j # 2

i # 1

"

, 1& i, j & n. (42)

The Pascal matrix Pn enjoys the following properties:

(i) Pn is totally positive;

(ii) Pn is positive definite;

(iii) The eigenvalues of the matrix Pn are real and positive;

(iv) If & (= 0 is an eigenvalue of Pn, then 1
& is also an eigenvalue of Pn;

(v) det(Pn) = 1;

(vi) The Cholesky’s factorization [2] of the matrix, Pn is given by:

Pn = AAT ,

where A is the Pascal matrix defined by:

A=

6
!

i # 1

j # 1

"
7n

i, j=1
; (43)

(vii) The explicit form of the inverse matrix, P#1
n =Qn = ('i j)

n
i, j=1 is given by [15]:

'i j = (#1)i+ j
n
#

k=max(i, j)

!

k# 1

i # 1

"!

k# 1

j # 1

"

; (44)

(viii) The matrix, Pn satisfies:

Pn = AB#1snV = T V, (45)

where T = (Ti j)
n
i, j=1 is an n" n lower triangular stochastic matrix given by:

Ti j =
1

(i # 1)!
c(i # 1, j # 1), 1& i, j & n, i % j, (46)

V =
:

ji#1
;n

i, j=1
, A is given in (43) and B = diag(0!, 1!, 2!, . . . , (n# 1)!) (for more details,

see [13, 14, 27, 28]);

(ix) The entries of the matrix, Pn satisfy the recurrence relation:

Pi1 = P1 j = 1, (47)

and

Pi j = Pi, j#1 + Pi#1, j . (48)

(x) Let Rn be the matrix obtained from the Pascal matrix Pn by subtracting one from the

element in position (n, n) of Pn, then det(Rn) = 0.



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 143

2. Main Results

This section is mainly devoted to study the n" n inverse matrix Qn, in (44), of the n" n

symmetric Pascal matrix Pn in (42). The main object is to find a recurrence relation satisfied

by the entries of Qn. Setting j = 1 in (44), we obtain:

'i1 = (#1)i+1

!

n

i

"

= '1i , 1& i & n, (49)

having used Lemma 1. Putting j = n in (44), yields:

'in = (#1)i+n

!

n# 1

i # 1

"

= 'ni , 1& i & n. (50)

At this point we may formulate the following result.

Theorem 1. The entries of the inverse matrix Qn = P#1
n = ('i j)

n
i, j=1, in (44), satisfy:

'i, j = 'i, j+1 + 'i+1, j + (#1)i+ j

!

n

i

"!

n

j

"

, 1& i, j & n. (51)

Proof. It is well-known that

!

m

n

"

=

!

m# 1

n# 1

"

+

!

m# 1

n

"

. (52)

From (44), we have

'i, j = (#1)i+ j
n
#

k=1

!

k# 1

i # 1

"!

k# 1

j # 1

"

. (53)

By using the identity (52), we get

'i, j =(#1)i+ j

6 n
#

k=1

<!

k

i

"

#

!

k# 1

i

"=<!

k

j

"

#

!

k# 1

j

"=
7

=(#1)i+ j

6 n
#

k=1

!

k

i

"!

k

j

"

#
n
#

k=1

!

k

i

"!

k# 1

j

"

#
n
#

k=1

!

k# 1

i

"!

k

j

"

+

n
#

k=1

!

k# 1

i

"!

k# 1

j

"
7

=(#1)i+ j

6
!

n

i

"!

n

j

"

+

n#1
#

k=1

!

k

i

"!

k

j

"

#
n
#

k=1

!

k

i

"!

k# 1

j

"

#
n
#

k=1

!

k# 1

i

"!

k

j

"

+

n
#

k=1

!

k# 1

i

"!

k# 1

j

"
7



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 144

=(#1)i+ j

6
!

n

i

"!

n

j

"

+

n
#

k=1

!

k# 1

i

"!

k# 1

j

"

#
n
#

k=1

!

k

i

"!

k# 1

j

"

#
n
#

k=1

!

k

j

"!

k# 1

i

"

+

n
#

k=1

!

k# 1

i

"!

k# 1

j

"
7

=(#1)i+ j

6
!

n

i

"!

n

j

"

#
n
#

k=1

!

k# 1

j

"<!

k

i

"

#

!

k# 1

i

"=

#
n
#

k=1

!

k# 1

i

"<!

k

j

"

#

!

k# 1

j

"=
7

=(#1)i+ j

6
!

n

i

"!

n

j

"

#
n
#

k=1

!

k# 1

j

"!

k# 1

i # 1

"

#
n
#

k=1

!

k# 1

i

"!

k# 1

j # 1

"
7

=(#1)i+ j

6
!

n

i

"!

n

j

"

+ (#1)i+ j'i, j+1 + (#1)i+ j'i+1, j

7

. (54)

Therefore

'i, j = 'i, j+1 + 'i+1, j + (#1)i+ j

!

n

i

"!

n

j

"

. (55)

Using (49), (50) and (55), we see that in order to obtain Qn, we only need to compute

the 1
2(n# 1)(n# 2) elements '22,'32, . . . ,'n#1,n#1 and taking into account the fact that Qn is

symmetric. Also, the solution of the linear system of the Pascal type

Pn[u1u2 . . . un]
T = [ f1 f2 . . . fn]

T (56)

may be obtained in O(n2) operations by using (49), (50) and (55). The following is a MAPLE

code to compute the inverse matrix of the Pascal matrix, Pn for n= 6, as an example.

MAPLE Code for Inverting Pascal Matrix.

> r e s t a r t :

> n:=6: Q:= array ( 1 . . n , 1 . . n , symmetric ) :

> f o r i to n do

Q[ i , n]:=(#1)^(n+ i )! binomial (n#1, i #1):

Q[ i ,1]:=(#1)^( i+1)!binomial (n , i ) :

od :

> f o r i from n#1 by #1 to 2 do

f o r j from i by #1 to 2 do

Q[ i , j ]:=Q[ i , j+1]+Q[ i+1, j ]+(#1)^( i+ j )! binomial (n , i )! binomial (n , j ) :

od :

od :

> Q:=evalm (Q) :

># Check using the l i n a l g package .

> P:= l i n a l g [ i nve r s e ] (Q) :



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 145

The results are printed as shwon.

Q =

%

&

&

&

&

&

&

'

6 #15 20 #15 6 #1

#15 55 #85 69 #29 5

20 #85 146 #127 56 #10

#15 69 #127 117 #54 10

6 #29 56 #54 26 #5

#1 5 #10 10 #5 1

(

)

)

)

)

)

)

*

P =

%

&

&

&

&

&

&

'

1 1 1 1 1 1

1 2 3 4 5 6

1 3 6 10 15 21

1 4 10 20 35 56

1 5 15 35 70 126

1 6 21 56 126 252

(

)

)

)

)

)

)

*

3. Applications

In this section we are going to give a new proof for El-Mikkawy conjecture [14]. Moreover,

some new identities will be obtained.

3.1. A New Proof for El-Mikkawy Conjecture [14]

Theorem 2. Let G be the n" n matrix whose (i, j) entry is given by "
(n)
i, j (x1, x2, . . . , xn) and let

V be the n" n Vandermonde matrix defined by (27). Then we have

det(G) = (#1)
n(n#1)

2 det(V ) =

$

det(V ) if n) 0 or 1 mod(4),

#det(V ) if n) 2 or 3 mod(4).
(57)

Proof. For i = 1,2, . . . , n, let Fi as given in (31), D and D̃ are n" n diagonal matrices given

by: D = diag( 1
F1

, 1
F2

, . . . , 1
Fn
) and D̃ = diag((#1)n#1, (#1)n#2, . . . , (#1)0).

It can be shown that for the matrices D, J and D̃, we have

det(D) =
1

n
5

i=1

Fi

=
1

:

(#1)
n(n#1)

2

6

det(V )

72

)

, (58)

having used (28) and (31),

det(J) = (#1)
n(n#1)

2 and det(D̃) = (#1)
n(n#1)

2 . (59)

By noticing that the matrix V#1 in (30) can be written in the form:

V#1 = DGJ D̃, (60)

the result follows.



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 146

3.2. New Identities

In [13] it is shown that the Pascal matrix, Pn and the Vandermonde matrix, V = ( ji#1)ni, j=1
are related by:

Pn = T V. (61)

From (61), we get:

Qn = ('i j)
n
i, j=1 = P#1

n = V#1T#1, (62)

where

T#1 = ((i j)
n
i, j=1, (63)

with

(i j = (#1)i+ j( j # 1)!S(i # 1, j # 1), 1& i, j & n, i % j. (64)

Thus

'i j =

n
#

k=1

%ik(k j =

n
#

k=1

6

(#1)i+k

(n# 1)!

!

n# 1

i # 1

"

"
(n)
n#k+1,i

76

(#1) j+k( j # 1)!S(k# 1, j # 1)

7

=(#1)i+ j ( j # 1)!

(i # 1)!(n# i)!

n
#

k=1

"
(n)
n#k+1,i

S(k# 1, j # 1).

(65)

Rewriting (44) in the form:

'i j = (#1)i+ j
n
#

k=1

!

k# 1

i # 1

"!

k# 1

j # 1

"

, (66)

then using (65) and (66), yields:

n
#

k=max (i, j)

!

k# 1

i # 1

"!

k# 1

j # 1

"

=
( j # 1)!

(i # 1)!(n# i)!

n
#

k= j

"
(n)
n#k+1,i

S(k# 1, j # 1). (67)

Setting i = n on both sides of (67) gives

!

n# 1

j # 1

"

=
( j # 1)!

(n# 1)!0!

n
#

k= j

"
(n)
n#k+1,n

S(k# 1, j # 1)

=
( j # 1)!

(n# 1)!

n
#

k= j

"
(n#1)
n#k

(1,2, . . . , n# 1)S(k# 1, j # 1)

=
( j # 1)!

(n# 1)!

n
#

k= j

c(n, k)S(k# 1, j # 1)

=
1

(n# j)!

!

n# 1

j # 1

"

n
#

k= j

c(n, k)S(k# 1, j # 1).

(68)



M. El-Mikkawy, F. Atlan / Eur. J. Pure Appl. Math, 8 (2015), 135-151 147

Hence

n
#

k= j

c(n, k)S(k# 1, j # 1) =(n# j)!

!

n# 1

j # 1

"2

=(n# 1)n# j

!

n# 1

n# j

"

. "

(69)

From (61), we see that

!

i + j # 2

i # 1

"

=

n
#

k=1

>

1

(i # 1)!
c(i # 1, k# 1)

?>

jk#1

?

.

Then
!

i + j # 2

i # 1

"

=

n
#

k=1

jk#1

(i # 1)!
c(i # 1, k# 1). (70)

Setting j = 2 in (70), we obtain

n
#

k=1

2k#1c(i # 1, k# 1) = (i # 1)!

!

i

i # 1

"

= i!. (71)

Therefore, for any positive integers n and 1& i & n, we have

n
#

k=1

2k#1c(i # 1, k# 1) = i!. " (72)

We list the following additional identities, without proof, for the sake of space require-

ments. In all cases n is a positive integer and i, j are integers such that 1& i, j & n.

•
n
/

r=1

(#1)r+1

!

i + r # 2

i # 1

"!

n

r

"

= !i1.

•
n
/

k=1

(#1)k+1ki#1

!

n

k

"

= !i1.

•
n
/

k=1

(#1)k+1"
(n#1)
n#k

(2,3, . . . , n) = (n# 1)!.

•
n
/

k=1

s(n, k) jk#1 = (n# 1)!!n j .

•
n
/

r= j

!

n# 1

n# r

"

(n# 1)n#r s(r # 1, j # 1) = c(n, j).

•
n
/

r= j

n
/

k=1

(#1) j+k

!

n# 1

k# 1

"

"
(n)
n#r+1,k

S(r # 1, j # 1) = 0.



REFERENCES 148

•
n
/

r=1

n
/

k=1

(#1)k
!

n# 1

k# 1

"!

i + k# 2

k# 1

"

"
(n)
n#r+1,k

=

+

,

-

,

.

#(n# 1)! if i = 1

(n# 1)! if i = 2

0 otherwise

.

•
n
/

r=1

n
/

k=1

(#1)kk2

!

n

k

"

"
(n)
n#r+1,k

= n!.

•
n
/

k=max(i, j)

@

A

k

i

B

C

k! c(k, j) =

@

A

n

i

B

C

n! "
(n#1)
n# j (1,2, . . . , i # 1, i + 1, . . . , n);

•
n
/

k=max(i, j)

(n# i)n#kc(k, j) = "
(n#1)
n# j (1,2, . . . , i # 1, i + 1, . . . , n);

•
n
/

k=i

@

A

n# i

n# k

B

C

@

A

n# 1

n# k

B

C

=
n
/

k=i

(n#i)n#k

(n#1)n#k
= n

i ;

•
n
/

k=i

@

A

k

i

B

C

k =
n
/

k=i

(k#1)i#1

i! =

@

A

n

i

B

C

i ;

•
n
/

k=i

(k)i
k =

(n)i
i .

•
n
/

k=1

ks(n+ 1, k+ 1) = (#1)n#1(n# 1)!= s(n, 1).

ACKNOWLEDGEMENTS The authors are extremely grateful to the editorial team of EJPAM

and anonymous referees for their useful comments and suggestions that have helped in im-

proving the readability of this paper.

References

[1] L. Aceto and D. Trigiante. The matrices of Pascal and other greats. American Mathematical

Monthly, 108:232–245, 2001.

[2] M. B. Allen and E. L. Isaacson. Numerical Analysis for Applied Science. John Wiley & Sons,

Hoboken, USA, 1997.

[3] R. Brawer and M. Pirovino. The linear algebra of the Pascal matrix. Linear Algebra and

its Applications, 174:13–23, 1992.



REFERENCES 149

[4] G. S. Call and D. J. Velleman. Pascal’s matrices. American Mathematical Monthly,

100(4):372–376, 1993.

[5] G. S. Cheon and M. El-Mikkawy. Extended symmetric Pascal matrices via hypergeometric

functions. Applied Mathematics and Computation, 158:159–168, 2004.

[6] G. S. Cheon and J. S. Kim. Stirling matrix via Pascal matrix. Linear Algebra and its

Applications, 329:49–59, 2001.

[7] G. S. Cheon and J. S. Kim. Factorial Stirling matrix and related combinatorial sequences.

Linear Algebra and its Applications, 357:247–258, 2002.

[8] J. W. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, 1997.

[9] A. Edelman and G. Strangl. Pascal matrices. American Mathematical Monthly, Cambridge,

2003.

[10] M. E. A. El-Mikkawy. An algorithm for solving Vandermonde systems. Journal of Institute

of Mathematics and Computer Science, 3(3):293–297, 1990.

[11] M. E. A. El-Mikkawy. Explicit inverse of a generalized Vandermonde matrix. Applied

Mathematics and Computation, 146:643–651, 2003.

[12] M. E. A. El-Mikkawy. On a connection between symmetric polynomials, generalized

Stirling numbers and the Newton general divided difference interpolation polynomial.

Applied Mathematics and Computation, 138(2):375–385, 2003.

[13] M. E. A. El-Mikkawy. On a connection between the Pascal, Vandermonde and Stirling

matrices-I. Applied Mathematics and Computation, 145(1):23–32, 2003.

[14] M. E. A. El-Mikkawy. On a connection between the Pascal, Vandermonde and Stirling

matrices-II. Applied Mathematics and Computation, 146(2):759–769, 2003.

[15] M. E. A. El-Mikkawy. On solving linear systems of the Pascal type. Applied Mathematics

and Computation, 136:195–202, 2003.

[16] M. E. A. El-Mikkawy and F. Atlan. Remarks on two symmetric polynomials and some

matrices. Applied Mathematics and Computation, 219:8770–8778, 2013.

[17] M. E. A. El-Mikkawy and T. Sogabe. Notes on particular symmetric polynomials with

applications. Applied Mathematics and Computation, 215:3311–3317, 2010.

[18] S. M. Fallat. Bidiagonal factorizations of totally nonnegative matrices. American Mathe-

matical Monthly, 108(8):697–712, 2001.

[19] T. X. He and J. S. Shiue. A note on Horner’s method. Journal of Concrete and Applicable

Mathematics, 10(1):53–64, 2012.

[20] J. G. Kemeny and J. L. Snell. Finite Markov chains. Springer-Verlag, New York, 1976.



REFERENCES 150

[21] X. G. Lv, T. Z. Huang, and Z. G. Ren. A new algorithm for linear systems of the Pascal

type. Journal of computational and applied mathematics, 225(1):309–315, 2009.

[22] C. Y. Ma and S. L. Yang. Pascal type matrices and Bernoulli numbers. International

Journal of Pure and Applied Mathematics, 58(3):249–254, 2010.

[23] E. N. Onwuchekwa. Some classes of lower triangular matrices and their inverses. Inter-

national Journal of Mathematical Sciences and Applications, 1(3):1169–1180, 2011.

[24] H. Oruc and H. K. Akmaz. Symmetric functions and the Vandermonde matrix. Journal

of Computational and Applied Mathematics, 172:49–64, 2004.

[25] T. Sogabe and M. E. A. El-Mikkawy. On a problem related to the Vandermonde determi-

nant. Discrete Applied Mathematics, 157:2997–2999, 2009.

[26] R. Vein and P. Dale. Determinants and their applications in mathematical physics. Springer,

New York, 1999.

[27] W. Wang and T. Wang. Commentary on an open question. Applied Mathematics and

Computation, 196(1):353–355, 2008.

[28] W. Wang and T. Wang. Remarks on two special matrices. Ars Combinatoria, 94:521–535,

2010.

[29] X. Wang. A Stable fast algorithm for solving linear systems of the Pascal type. Journal of

Computational Analysis and Applications, 9(4):411–419, 2007.

[30] X. Wang and L. Lu. A fast algorithm for solving linear systems of the Pascal type. Applied

Mathematics and Computation, 175(1):441–451, 2006.

[31] S. L. Yang. On the LU factorization of the Vandermonde matrix. Discrete Applied Mathe-

matics, 146:102–105, 2005.

[32] S. L. Yang. On a connection between the Pascal, Stirling and Vandermonde matrices.

Discrete Applied Mathematics, 155(15):2025–2030, 2007.

[33] S. L. Yang and Z. K. Qiao. The Bessel numbers and Bessel matrices. Journal of Mathe-

matical Research and Exposition, 31(4):627–636, 2011.

[34] S. L. Yang and H. You. On a relationship between Pascal matrix and Vandermonde matrix.

Journal of Mathematical Research and Exposition, 26(1):33–39, 2006.

[35] Z. Zeng, Y. Tu, and J. Xiao. A neural-network method based on RLS algorithm for solv-

ing special linear systems of equations. Journal of Computational Information Systems,

8(7):2915–2920, 2012.

[36] Z. Z. Zhang. The linear algebra of the generalized Pascal matrix. Linear Algebra and its

Applications, 250:51–60, 1997.



REFERENCES 151

[37] Z. Z. Zhang and M. Liu. An extension of the generalized Pascal matrix and its algebraic

properties. Linear Algebra and its Applications, 271:169–177, 1998.