Remarks on two symmetric polynomials and some matrices
Applied Mathematics and Computation 219 (2013) 8770–8778
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Remarks on two symmetric polynomials and some matrices Moawwad El-Mikkawy ⇑, Faiz Atlan Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
a b s t r a c t By means of complete symmetric polynomials this paper gives a new proof for the Vandermonde determinant formula. Another alternative proof for this formula is obtained via the collocation matrices. It also gives a generalized relationship between the Vandermonde, the Pascal and the Stirling matrices. A new approach to obtain the explicit inverse of the Vandermonde matrix is investigated. Closed form expressions for some Vandermonde related determinants are obtained. Published by Elsevier Inc.
Keywords: Interpolation Stirling numbers Vandermonde matrix Symmetric polynomials Pascal matrix
1. Introduction and Preliminaries Symmetric polynomials are widely used in many fields such as numerical analysis [6,9,26,29], algebra [2,4,16,17], combinatorics [1,5,8], statistics [21,25], physics [23], discrete mathematics [27], and many others. It is of prime importance to explore new properties of the symmetric polynomials. For the basic properties of symmetric polynomials, see [24]. There are many special types of matrices which are of great importance in many scientific and engineering work. For instance matrices of type Pascal [15], and of type Vandermonde [13,28]. This paper is mainly concerned with matrices of Vandermonde, Pascal and Stirling types. The Vandermonde type matrices, for instance, frequently appear in many applications. Among them curve fitting, interpolation, scattering and in the derivation of explicit Runge–Kutta and Runge–Kutta-Nystrom numerical methods [11,12,14]. The current paper is organized as follows: In the next section, we describe a new proof for the Vandermonde determinant formula using the complete symmetric polynomials. In Section 3, we give a new approach for the explicit inverse of the Vandermonde matrix. A generalized relationship between the Vandermonde, the Pascal and the Stirling matrices is given. Finally, closed form expressions for some Vandermonde related determinants are obtained in Section 4. The rest of this section gives some basic definitions. Throughout this paper x denotes the set fx1 ; x2 ; . . . ; xn g. Definition 1.1 [19]. The Vandermonde matrix VðxÞ is a matrix of the form
n VðxÞ ¼ xi1 j
i;j¼1
ð1Þ
:
The Vandermonde determinant formula is well known, see for instance [29], in many text books and articles. It is given by
n det VðxÞ ¼ xji1
i;j¼1
¼
Y
ðxi xj Þ:
16j > < 1 eðnÞ r ðxÞ :¼ > > :
if r > n or n < 0; if r ¼ 0;
X
xi 1 xi 2 . . . xi r
if r ¼ 1; 2; . . . n
ð16Þ
16i1 > :
if r < 0 or n < 0 or ðn ¼ 0 and r > 0Þ; if r ¼ 0;
X
xi1 xi2 . . . xir
16i1 6i2 6...6ir 6n ðnÞ
It should be noticed that each er has written as
X
eðnÞ r ðxÞ :¼
ð17Þ
if r ¼ 1; 2; . . .
n nþr1 ðnÞ terms and each hr has terms. Moreover these polynomials can be r r
xk11 xk22 . . . xknn
r ¼ 0; 1; . . . ; n
ð18Þ
k1 þ k2 þ þ kn ¼ r k1 ; k2 ; . . . ; kn 2 f0; 1g and
X
ðnÞ
hr ðxÞ :¼
xd11 xd22 . . . xdnn
r ¼ 0; 1; . . .
ð19Þ
d1 þ d2 þ þ dn ¼ r d1 ; d2 ; . . . ; dn 2 f0; 1; . . . ; rg ðnÞ
The generating functions for er
EðtÞ ¼
n Y
ð1 þ xi tÞ ¼
ðnÞ
and hr
are given, respectively by
n X
r eðnÞ r t ;
ð20Þ
r¼0
i¼1
and
HðtÞ ¼
n Y
ð1 xi tÞ1 ¼
1 X ðnÞ hr t r :
ð21Þ
r¼0
i¼1
From (20) we see that n n X X r eðnÞ erðn1Þ t r : r t ¼ ð1 þ xn tÞ r¼0
ð22Þ
r¼0
Comparing the coefficients of tr on both sides of (22) yields the following recurrence relation ðn1Þ
ðn1Þ eðnÞ ðx1 ; x2 ; . . . ; xn1 Þ þ xn er1 ðx1 ; x2 ; . . . ; xn1 Þ; r ðx1 ; x2 ; . . . ; xn Þ ¼ er
ð23Þ
The falling factorial ðxÞn in (7) can be written in the form
ðxÞn ¼
n X k¼0
ð1Þnk enk ð0; 1; . . . ; n 1Þxk : ðnÞ
ð24Þ
M. El-Mikkawy, F. Atlan / Applied Mathematics and Computation 219 (2013) 8770–8778
8773
Comparing the coefficients of xk in (5) and (24), yields ðnÞ
ðn1Þ
cðn; kÞ ¼ enk ð0; 1; . . . ; n 1Þ ¼ enk ð1; 2; . . . ; n 1Þ:
ð25Þ
It can also be shown that ðkÞ
Sðn; kÞ ¼ hnk ð1; 2; . . . ; kÞ:
ð26Þ
For any xj 2 fx1 ; x2 ; . . . ; xn g, we have ðnÞ
ðn1Þ
ei ðx1 ; x2 ; . . . ; xn Þ ¼ ei
ðn1Þ
ðx1 ; x2 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þ þ xj ei1 ðx1 ; x2 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þ:
ð27Þ
Partial differentiation for both sides of (27) with respect to xj gives
@ ðnÞ ðnÞ ðn1Þ e ¼ ei;j ¼ ei1 ðx1 ; x2 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þ: @xj i
ð28Þ
Therefore by using (16), we have
ðnÞ
eij ðxÞ ¼
8 0 > > > > > n; if i ¼ 1;
X
> > > 1 6 r 1 < r 2 < < r i1 6 n > > : r 1 ; r2 . . . ; r i1 –j
xr1 xr2 . . . xri1
if i ¼ 2; 3; . . . n:
ð29Þ
Let G be the n n matrix defined by G ¼ ðeij Þni;j¼1 . Then (see [19,27]) ðnÞ
det G ¼
det VðxÞ
if n 0 or 1 mod ð4Þ;
ð30Þ
det VðxÞ if n 2 or 3 mod ð4Þ:
Definition 1.8 [6]. Given a set of n data points relating a dependent variable y ¼ f ðxÞ to an independent variable x as follows
x
x1
x2
….
xn
f(x)
f( x1 )
f( x 2 )
….
f( x n )
The zeroth order divided difference f ½xi is defined as follows
f ½xi ¼ f ðxi Þ;
i ¼ 1; 2; . . . ; n:
ð31Þ
Higher order divided differences are given recursively by
f ½x1 ; x2 ; . . . ; xk1 ; xk ¼
f ½x2 ; x3 ; . . . ; xk f ½x1 ; x2 ; . . . ; xk1 ; xk x 1
k ¼ 2; 3; . . . ; n:
ð32Þ
It is known that the divided difference f ½x1 ; x2 ; . . . ; xn satisfies [26]
f ½x1 ; x2 ; . . . ; xn ¼
n X i¼1
f ðxi Þ n Y
:
ð33Þ
ðxi xr Þ
r¼1 r–i Therefore it is a symmetric function of its arguments x1 ; x2 ; . . . ; xn . ðnÞ From [26], for any integer k, the complete symmetric polynomial hk satisfies ðkÞ
hnk ð1; 2; . . . ; kÞ ¼ f ½0; 1; . . . ; k:
ð34Þ
n
where f ðxÞ ¼ x . In conclusion, we have ðkÞ
Sðn; kÞ ¼ hnk ð1; 2; . . . ; kÞ ¼ f ½0; 1; . . . ; k;
ð35Þ
having used (26) and (34), where f ðxÞ ¼ xn . 2. A new proof for the Vandermonde determinant formula This section is mainly devoted to give a new proof for the Vandermonde determinant formula given by (2). The proof is based on using the complete symmetric polynomials.
8774
M. El-Mikkawy, F. Atlan / Applied Mathematics and Computation 219 (2013) 8770–8778
We begin this section by giving the following result. ðnÞ
Lemma 2.1. The complete symmetric polynomial hr ðx1 ; x2 ; . . . ; xn Þ satisfies ðn1Þ
hr
ðn1Þ
ðx2 ; x3 ; . . . ; xn Þ hr
ðnÞ
ðx1 ; x2 ; . . . ; xn1 Þ ¼ ðxn x1 Þhr1 ðx1 ; x2 ; . . . ; xn Þ:
ð36Þ
Proof. The generating function (21) gives 1 1 X X ðnÞ ðn1Þ r hr ð1 xn tÞt r ¼ hr t: r¼0
ð37Þ
r¼0
Comparing the coefficients of t r on both sides of (37), yields the following recurrence relation ðnÞ
ðn1Þ
hr ðx1 ; x2 ; . . . ; xn Þ ¼ hr
ðnÞ
ð38Þ
ðnÞ
ð39Þ
ðx1 ; x2 ; . . . ; xn1 Þ þ xn hr1 ðx1 ; x2 ; . . . ; xn Þ:
Substituting x1 $ xn , in (38) we get ðnÞ
ðn1Þ
hr ðx1 ; x2 ; . . . ; xn Þ ¼ hr
ðx1 ; x2 ; . . . ; xn1 Þ þ x1 hr1 ðx1 ; x2 ; . . . ; xn Þ:
From (38) and (39) by subtraction we obtain the identity (36). h At this point it is convenient to define
PðkÞ r ¼
k Y
ðxiþr xi Þ;
1 6 r; k 6 n 1
ð40Þ
i¼1
and
nkþ1 ðkÞ W nkþ1 ¼ hi1 ðxj ; xjþ1 ; . . . ; xjþk1 Þ ;
k ¼ 1; 2; . . . ; n:
i;j¼1
ð41Þ
On setting k ¼ 1 in (41), we get
n ð1Þ W n ¼ hi1 ðxj Þ
i;j¼1
n ¼ xji1
i;j¼1
¼ det VðxÞ:
ð42Þ
Putting k ¼ n in (41), we have ðnÞ
W 1 ¼ h0 ðx1 ; x2 ; . . . ; xn Þ ¼ 1
ð43Þ
having used (17). For k ¼ 1; 2; . . . ; n 1, in this order, perform the column operations C r C r1 ; r ¼ 2; 3; . . . ; n k þ 1, on W nkþ1 and taking into account the identity (36), we obtain ðnkÞ
W nkþ1 ¼ Pk
W nk ;
k ¼ 1; 2; . . . ; n 1:
ð44Þ
From (44) we see that ðn1Þ ðn2Þ P2
W n ¼ det VðxÞ ¼ P1
ð1Þ
ðn1Þ ðn2Þ P2
. . . Pn1 W 1 ¼ P1
Y
ð1Þ
. . . Pn1 ¼
ðxi xj Þ:
ð45Þ
16j j. In conclusion
det VðxÞ ¼ det V Q ðxÞ ¼
n Y
Y
qr ðxr Þ ¼
r¼1
ðxi xj Þ:
16j > < 16j ðxi xj Þðt i t j Þ for n 2 or 3 mod ð4Þ > : 16j < xj > :
xij
if i ¼ 1; 2; . . . ; k; k ¼ 0; 1; 2; . . . ; n;
ð69Þ
if i ¼ k þ 1; k þ 2; . . . ; n:
Then, we have ðnÞ
det F n ðxÞ ¼ hr ðx1 ; x2 ; . . . ; xn Þ: det VðxÞ;
ð70Þ
det Gn ðxÞ ¼ f ½x0 ; x1 ; . . . ; xn : det VðxÞ;
ð71Þ
det Hn ðxÞ ¼ ð1Þn1 : det VðxÞ;
ð72Þ
det X n ðxÞ ¼ ð1Þn1 : det VðxÞ; and
ð73Þ
ðnÞ
det V k ðxÞ ¼ enk : det VðxÞ:
ð74Þ
Notice that by setting k ¼ n in (70) we get
V n ðxÞ ¼ VðxÞ:
ð75Þ
Also notice that for the case k ¼ n we see that (75), as expected, yields
det V n ðxÞ ¼ det VðxÞ;
ð76Þ
having used (16). References [1] E.E. Allen, The descent monomials and a basis for the diagonally symmetric polynomials, J. Alg. Combin. 3 (1994) 5–16. [2] K.M. Andersen, A characterization of polynomials, Math. Mag. 69 (1996) 137–142. [3] T. Bella, Y. Eidelmany, I. Gohbergy, V. Olshevsky, Computations with quasiseparable polynomials and matrices, Theor. Comput. Sci. 409 (2008) 158– 179. [4] C. Bertone, The Euler characteristic as a polynomial in the Chern classes, Int. J. Algebra 2 (2008) 757–769. [5] T. Bickel, N. Galli, K. Simon, Birth processes and symmetric polynomials, Ann. Combin. 5 (2001) 123–139. [6] R.L. Burden, J.D. Faires, Numerical Analysis, Seventh ed., Brooks & Cole, 2001. [7] G.S. Cheon, J.S. Kim, Stirling matrix via Pascal matrix, Linear Algebra Appl. 329 (2001) 49–59. [8] W.Y.C. Chen, C. Krattenthaler, A.L.B. Yang, The flagged Cauchy determinant, Graphs & Combin. 21 (2005) 51–62. [9] J.W. Demmel, Numerical Linear Algebra, SIAM Publications, Philadelphia, PA, 1996. [10] J. Demmel, P. Koev, Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials, Linear Algebra Appl. 417 (2006) 382–396. [11] J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of embedded Runge–Kutta–Nystrom formulae, IMA J. Numer. Anal. 7 (1987) 235–250. [12] J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, High order embedded Runge–Kutta–Nystrom formulae, IMA J. Numer. Anal. 7 (1987) 423–430. [13] M.E.A. El-Mikkawy, An algorithm for solving Vandermonde systems, J. Inst. Math. Comput. Sci. 3 (3) (1990) 293–297. [14] M.E.A. El-Mikkawy, A better approach for the derivation of the embedded Runge–Kutta–Nystrom pair of orders 6 and 4, Int. J. Comput. Math. 40 (1991) 245–249. [15] M.E.A. El-Mikkawy, On solving linear systems of the Pascal type, Appl. Math. Comput. 136 (2003) 195–202. [16] M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices-I, Appl. Math. Comput. 145 (2003) 23–32. [17] M.E.A. El-Mikkawy, Explicit inverse of a generalized Vandermonde matrix, Appl. Math. Comput. 146 (2003) 643–651. [18] M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices-II, Appl. Math. Comput. 146 (2003) 759–769. [19] M.E.A. El-Mikkawy, T. Sogabe, Notes on particular symmetric polynomials with applications, Appl. Math. Comput. 215 (2010) 3311–3317. [20] W. Gander, Change of basis in polynomial interpolation, Numer. Linear Algebra Appl. 12 (2005) 769–778. [21] D.L. Hydorn, R.J. Muirhead, Polynomial estimation of eigenvalues, Commun. Stat. Theory Methods 28 (1999) 581–596. [22] John G. Kemeny, J. Laurie Snell, Finite Markov Chains, Springer-Verlag, New York, 1976. [23] S.V. Lyudkovskii, Compact relationships between invariants of classical Lie groups and elementary symmetric polynomials, Theor. Math. Phys. 89 (1991) 1281–1286. [24] I.G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford University Press, Oxford, 1995. [25] P. Major, The limit behavior of elementary symmetric polynomials of I.I.D. random variables when their order tends to infinity, Ann. Probab. 27 (1999) 1980–2010. [26] G.M. Phillips, Interpolation and Approximation by Polynomials, Springer, Verlag, New York, 2003. [27] T. Sogabe, M.E.A. El-Mikkawy, On a problem related to the Vandermonde determinant, Discrete Appl. Math. 157 (2009) 2997–2999. [28] W.P. Tang, G.H. Golub, The block decomposition of a Vandermonde matrix and its applications, BIT 21 (1981) 505–517. [29] R. Vein, P. Dale, Determinants and their Applications in Mathematical Physics, Springer, New York, 1999. [30] W. Wang, T. Wang, Commentary on an open question, Appl. Math. Comput. 196 (1) (2008) 353–355. [31] Sheng-liang Yang, On the LU factorization of the Vandermonde matrix, Discrete Appl. Math. 146 (2005) 102–105.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Remarks on two symmetric polynomials and some matrices Moawwad El-Mikkawy ⇑, Faiz Atlan Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
a b s t r a c t By means of complete symmetric polynomials this paper gives a new proof for the Vandermonde determinant formula. Another alternative proof for this formula is obtained via the collocation matrices. It also gives a generalized relationship between the Vandermonde, the Pascal and the Stirling matrices. A new approach to obtain the explicit inverse of the Vandermonde matrix is investigated. Closed form expressions for some Vandermonde related determinants are obtained. Published by Elsevier Inc.
Keywords: Interpolation Stirling numbers Vandermonde matrix Symmetric polynomials Pascal matrix
1. Introduction and Preliminaries Symmetric polynomials are widely used in many fields such as numerical analysis [6,9,26,29], algebra [2,4,16,17], combinatorics [1,5,8], statistics [21,25], physics [23], discrete mathematics [27], and many others. It is of prime importance to explore new properties of the symmetric polynomials. For the basic properties of symmetric polynomials, see [24]. There are many special types of matrices which are of great importance in many scientific and engineering work. For instance matrices of type Pascal [15], and of type Vandermonde [13,28]. This paper is mainly concerned with matrices of Vandermonde, Pascal and Stirling types. The Vandermonde type matrices, for instance, frequently appear in many applications. Among them curve fitting, interpolation, scattering and in the derivation of explicit Runge–Kutta and Runge–Kutta-Nystrom numerical methods [11,12,14]. The current paper is organized as follows: In the next section, we describe a new proof for the Vandermonde determinant formula using the complete symmetric polynomials. In Section 3, we give a new approach for the explicit inverse of the Vandermonde matrix. A generalized relationship between the Vandermonde, the Pascal and the Stirling matrices is given. Finally, closed form expressions for some Vandermonde related determinants are obtained in Section 4. The rest of this section gives some basic definitions. Throughout this paper x denotes the set fx1 ; x2 ; . . . ; xn g. Definition 1.1 [19]. The Vandermonde matrix VðxÞ is a matrix of the form
n VðxÞ ¼ xi1 j
i;j¼1
ð1Þ
:
The Vandermonde determinant formula is well known, see for instance [29], in many text books and articles. It is given by
n det VðxÞ ¼ xji1
i;j¼1
¼
Y
ðxi xj Þ:
16j > < 1 eðnÞ r ðxÞ :¼ > > :
if r > n or n < 0; if r ¼ 0;
X
xi 1 xi 2 . . . xi r
if r ¼ 1; 2; . . . n
ð16Þ
16i1 > :
if r < 0 or n < 0 or ðn ¼ 0 and r > 0Þ; if r ¼ 0;
X
xi1 xi2 . . . xir
16i1 6i2 6...6ir 6n ðnÞ
It should be noticed that each er has written as
X
eðnÞ r ðxÞ :¼
ð17Þ
if r ¼ 1; 2; . . .
n nþr1 ðnÞ terms and each hr has terms. Moreover these polynomials can be r r
xk11 xk22 . . . xknn
r ¼ 0; 1; . . . ; n
ð18Þ
k1 þ k2 þ þ kn ¼ r k1 ; k2 ; . . . ; kn 2 f0; 1g and
X
ðnÞ
hr ðxÞ :¼
xd11 xd22 . . . xdnn
r ¼ 0; 1; . . .
ð19Þ
d1 þ d2 þ þ dn ¼ r d1 ; d2 ; . . . ; dn 2 f0; 1; . . . ; rg ðnÞ
The generating functions for er
EðtÞ ¼
n Y
ð1 þ xi tÞ ¼
ðnÞ
and hr
are given, respectively by
n X
r eðnÞ r t ;
ð20Þ
r¼0
i¼1
and
HðtÞ ¼
n Y
ð1 xi tÞ1 ¼
1 X ðnÞ hr t r :
ð21Þ
r¼0
i¼1
From (20) we see that n n X X r eðnÞ erðn1Þ t r : r t ¼ ð1 þ xn tÞ r¼0
ð22Þ
r¼0
Comparing the coefficients of tr on both sides of (22) yields the following recurrence relation ðn1Þ
ðn1Þ eðnÞ ðx1 ; x2 ; . . . ; xn1 Þ þ xn er1 ðx1 ; x2 ; . . . ; xn1 Þ; r ðx1 ; x2 ; . . . ; xn Þ ¼ er
ð23Þ
The falling factorial ðxÞn in (7) can be written in the form
ðxÞn ¼
n X k¼0
ð1Þnk enk ð0; 1; . . . ; n 1Þxk : ðnÞ
ð24Þ
M. El-Mikkawy, F. Atlan / Applied Mathematics and Computation 219 (2013) 8770–8778
8773
Comparing the coefficients of xk in (5) and (24), yields ðnÞ
ðn1Þ
cðn; kÞ ¼ enk ð0; 1; . . . ; n 1Þ ¼ enk ð1; 2; . . . ; n 1Þ:
ð25Þ
It can also be shown that ðkÞ
Sðn; kÞ ¼ hnk ð1; 2; . . . ; kÞ:
ð26Þ
For any xj 2 fx1 ; x2 ; . . . ; xn g, we have ðnÞ
ðn1Þ
ei ðx1 ; x2 ; . . . ; xn Þ ¼ ei
ðn1Þ
ðx1 ; x2 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þ þ xj ei1 ðx1 ; x2 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þ:
ð27Þ
Partial differentiation for both sides of (27) with respect to xj gives
@ ðnÞ ðnÞ ðn1Þ e ¼ ei;j ¼ ei1 ðx1 ; x2 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þ: @xj i
ð28Þ
Therefore by using (16), we have
ðnÞ
eij ðxÞ ¼
8 0 > > > > > n; if i ¼ 1;
X
> > > 1 6 r 1 < r 2 < < r i1 6 n > > : r 1 ; r2 . . . ; r i1 –j
xr1 xr2 . . . xri1
if i ¼ 2; 3; . . . n:
ð29Þ
Let G be the n n matrix defined by G ¼ ðeij Þni;j¼1 . Then (see [19,27]) ðnÞ
det G ¼
det VðxÞ
if n 0 or 1 mod ð4Þ;
ð30Þ
det VðxÞ if n 2 or 3 mod ð4Þ:
Definition 1.8 [6]. Given a set of n data points relating a dependent variable y ¼ f ðxÞ to an independent variable x as follows
x
x1
x2
….
xn
f(x)
f( x1 )
f( x 2 )
….
f( x n )
The zeroth order divided difference f ½xi is defined as follows
f ½xi ¼ f ðxi Þ;
i ¼ 1; 2; . . . ; n:
ð31Þ
Higher order divided differences are given recursively by
f ½x1 ; x2 ; . . . ; xk1 ; xk ¼
f ½x2 ; x3 ; . . . ; xk f ½x1 ; x2 ; . . . ; xk1 ; xk x 1
k ¼ 2; 3; . . . ; n:
ð32Þ
It is known that the divided difference f ½x1 ; x2 ; . . . ; xn satisfies [26]
f ½x1 ; x2 ; . . . ; xn ¼
n X i¼1
f ðxi Þ n Y
:
ð33Þ
ðxi xr Þ
r¼1 r–i Therefore it is a symmetric function of its arguments x1 ; x2 ; . . . ; xn . ðnÞ From [26], for any integer k, the complete symmetric polynomial hk satisfies ðkÞ
hnk ð1; 2; . . . ; kÞ ¼ f ½0; 1; . . . ; k:
ð34Þ
n
where f ðxÞ ¼ x . In conclusion, we have ðkÞ
Sðn; kÞ ¼ hnk ð1; 2; . . . ; kÞ ¼ f ½0; 1; . . . ; k;
ð35Þ
having used (26) and (34), where f ðxÞ ¼ xn . 2. A new proof for the Vandermonde determinant formula This section is mainly devoted to give a new proof for the Vandermonde determinant formula given by (2). The proof is based on using the complete symmetric polynomials.
8774
M. El-Mikkawy, F. Atlan / Applied Mathematics and Computation 219 (2013) 8770–8778
We begin this section by giving the following result. ðnÞ
Lemma 2.1. The complete symmetric polynomial hr ðx1 ; x2 ; . . . ; xn Þ satisfies ðn1Þ
hr
ðn1Þ
ðx2 ; x3 ; . . . ; xn Þ hr
ðnÞ
ðx1 ; x2 ; . . . ; xn1 Þ ¼ ðxn x1 Þhr1 ðx1 ; x2 ; . . . ; xn Þ:
ð36Þ
Proof. The generating function (21) gives 1 1 X X ðnÞ ðn1Þ r hr ð1 xn tÞt r ¼ hr t: r¼0
ð37Þ
r¼0
Comparing the coefficients of t r on both sides of (37), yields the following recurrence relation ðnÞ
ðn1Þ
hr ðx1 ; x2 ; . . . ; xn Þ ¼ hr
ðnÞ
ð38Þ
ðnÞ
ð39Þ
ðx1 ; x2 ; . . . ; xn1 Þ þ xn hr1 ðx1 ; x2 ; . . . ; xn Þ:
Substituting x1 $ xn , in (38) we get ðnÞ
ðn1Þ
hr ðx1 ; x2 ; . . . ; xn Þ ¼ hr
ðx1 ; x2 ; . . . ; xn1 Þ þ x1 hr1 ðx1 ; x2 ; . . . ; xn Þ:
From (38) and (39) by subtraction we obtain the identity (36). h At this point it is convenient to define
PðkÞ r ¼
k Y
ðxiþr xi Þ;
1 6 r; k 6 n 1
ð40Þ
i¼1
and
nkþ1 ðkÞ W nkþ1 ¼ hi1 ðxj ; xjþ1 ; . . . ; xjþk1 Þ ;
k ¼ 1; 2; . . . ; n:
i;j¼1
ð41Þ
On setting k ¼ 1 in (41), we get
n ð1Þ W n ¼ hi1 ðxj Þ
i;j¼1
n ¼ xji1
i;j¼1
¼ det VðxÞ:
ð42Þ
Putting k ¼ n in (41), we have ðnÞ
W 1 ¼ h0 ðx1 ; x2 ; . . . ; xn Þ ¼ 1
ð43Þ
having used (17). For k ¼ 1; 2; . . . ; n 1, in this order, perform the column operations C r C r1 ; r ¼ 2; 3; . . . ; n k þ 1, on W nkþ1 and taking into account the identity (36), we obtain ðnkÞ
W nkþ1 ¼ Pk
W nk ;
k ¼ 1; 2; . . . ; n 1:
ð44Þ
From (44) we see that ðn1Þ ðn2Þ P2
W n ¼ det VðxÞ ¼ P1
ð1Þ
ðn1Þ ðn2Þ P2
. . . Pn1 W 1 ¼ P1
Y
ð1Þ
. . . Pn1 ¼
ðxi xj Þ:
ð45Þ
16j j. In conclusion
det VðxÞ ¼ det V Q ðxÞ ¼
n Y
Y
qr ðxr Þ ¼
r¼1
ðxi xj Þ:
16j > < 16j ðxi xj Þðt i t j Þ for n 2 or 3 mod ð4Þ > : 16j < xj > :
xij
if i ¼ 1; 2; . . . ; k; k ¼ 0; 1; 2; . . . ; n;
ð69Þ
if i ¼ k þ 1; k þ 2; . . . ; n:
Then, we have ðnÞ
det F n ðxÞ ¼ hr ðx1 ; x2 ; . . . ; xn Þ: det VðxÞ;
ð70Þ
det Gn ðxÞ ¼ f ½x0 ; x1 ; . . . ; xn : det VðxÞ;
ð71Þ
det Hn ðxÞ ¼ ð1Þn1 : det VðxÞ;
ð72Þ
det X n ðxÞ ¼ ð1Þn1 : det VðxÞ; and
ð73Þ
ðnÞ
det V k ðxÞ ¼ enk : det VðxÞ:
ð74Þ
Notice that by setting k ¼ n in (70) we get
V n ðxÞ ¼ VðxÞ:
ð75Þ
Also notice that for the case k ¼ n we see that (75), as expected, yields
det V n ðxÞ ¼ det VðxÞ;
ð76Þ
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