Generalized Leibniz Functional Matrices and Divided Difference Form of the Lagrange–Bürmann Formula ∗ Peipei Tang a , Aimin Xu B

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Linear Algebra and its Applications 436 (2012) 618–630

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Linear Algebra and its Applications

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Generalized Leibniz functional matrices and divided difference form of the Lagrange–Bürmann formula ∗ Peipei Tang a , Aimin Xu b,

a Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, PR China b Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, PR China

ARTICLE INFO ABSTRACT

Article history: In this paper, we derive two versions of divided difference form of the Received 12 May 2011 Lagrange–Bürmann expansion formula. The first version expresses Accepted 14 July 2011 all coefficients by a form of a determinant. By means of the factor- Availableonline30August2011 izations of the generalized Leibniz functional matrices with respect Submitted by R.A. Brualdi to a sequence, we obtain the second version with coefficients represented by a finite sum. Some new matrix inversions are also inves- AMS classification: 05A15 tigated. 15A23 © 2011 Elsevier Inc. All rights reserved.

Keywords: Lagrange–Bürmann expansion formula Generalized Leibniz functional matrix Divided difference

1. Introduction

In mathematics, particularly matrix theory and combinatorics, the Pascal matrix has played a fundamental role. The Pascal matrix is a matrix containing the binomial coefﬁcients as its elements and it is deﬁned as follows ⎧ ⎨⎪ i , j if i j (P ) = n ij ⎩⎪ 0otherwise.

∗ Corresponding author. E-mail addresses: [email protected] (P. Tang), [email protected] (A. Xu).

The generalized Pascal matrix was deﬁned in [5]as ⎧ ⎨ − i xi j if i j, ( [ ]) = j Pn x ij ⎩ 0otherwise.

In recent years, wide concerns were on the Pascal matrix and its generalized forms [2,4,5,14,19–21]. Since one of the first formulas to study the connections between Stirling matrices and Pascal and Vandermonde matrices was obtained by Cheon and Kim [7], many remarkable results related to the factorizations of lower triangular matrices were obtained, for example, the Stirling matrices of the first kind and of the second kind [8], the Lah matrix [16], as well as the Leibniz functional matrix which was defined in [11]as ⎧ ( − ) ⎨ f i j (x) , (i−j)! if i j (Ln[f (x)])ij = ⎩ 0otherwise,

( ) where f k (x) stands for the kth order derivative of f (x).In[18], the authors introduced generalized Leibniz functional matrices and obtained the factorizations of the generalized Leibniz functional matrices. By those results they also derived serval novel factorization forms of some well-known matrices, such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix due to Spivey and Zimmer [15]. Furthermore, making use of factorization and inverse of Leibniz functional matrix, Ma [13] established an extension formula of the classical Lagrange–Bürmann expansion formula and also obtained some new matrix inversions. The work of extension of classical Lagrange– Bürmann expansion has been dated from last century. In 1973, Carlitz [6]firstdiscoveredaq-analogue of the Lagrange–Bürmann expansion formula by expressing any formal power series f (x) in terms of rational functions. Furthermore, Andrews [1]gaveamoregeneralq-analogue of Lagrange–Bürmann expansion formula and Liu [12] successively generalized Carlitz’s fundamental expansion formula to the q-analogue of the Taylor formula with an extra parameter. Divided differences as the coefficients in a Newton form arise in numerical analysis. There are wide applications in the study of spline interpolation and polynomial interpolation. Given a function f ,for distinct points x0, x1,...,xn, the divided differences of f are defined recursively as

[x0]f = f (x0), [x0, x1,...,xn−1]f −[x1, x2,...,xn]f [x0, x1,...,xn]f = , n 1. x0 − xn

By extending the deﬁnition for [x0, x1,...,xn]f inthecaseofdistinctarguments,wehaveasimilar formula for x0 x1 ··· xn as follows ⎧ ⎨ [ , ,..., ] −[ , ,..., ] x0 x1 xn−1 f x1 x2 xn f = , x −x if xn x0 [x0, x1,...,xn]f = (n)( ) 0 n ⎩ f x0 = . n! if xn x0 For the basic properties of divided differences, one can refer to the recent reference [3]. Here let us introduce a key formula for divided differences which is called the Steffensen formula:

Lemma 1.1 [3]. Let h(x) = f (x)g(x). If f and g are sufﬁciently smooth functions, then for arbitrary nodes x0, x1,...,xn,wehave n [x0, x1,...,xn]h = [x0, x1,...,xk]f [xk, xk+1,...,xn]g. k=0

If x0 = x1 =···= xn = x, then the Leibniz formula holds, namely,

n ( ) n ( ) ( − ) h n (x) = f k (x)g n k (x). k=0 k 620 P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630

From the properties of divided differences, one can easily understand the derivatives as the limit of the divided differences. That is to say, divided differences may be viewed as discrete derivatives. This gives a motivation to present matrix factorizations for the divided differences of the functions, which include some recent interesting identities in [18] as special cases. Then we give two versions of divided difference form of the Lagrange–Bürmann formula. In order to describe our results, we ﬁrst show some basic notations in [10]. The q-shifted factorials are deﬁned by

n −1 ∞ ( ; ) = , ( ; ) = ( − k), ( ; ) = ( − k). a q 0 1 a q n 1 aq a q ∞ 1 aq k=0 k=0 The q-binomial coefﬁcient is deﬁned by ⎡ ⎤ ( ; ) ⎣ n ⎦ = q q n . k (q; q)k(q; q)n−k

It is easy to show that ⎡ ⎤ ( −n; ) ⎣ n ⎦ k nk−(k) q q k = (−1) q 2 . k (q; q)k

Again for fixed parameter q,letDq,x be the q-derivative operator defined by ( ) − ( ) f x f qx n n−1 Dq,xf (x) := and D , f = Dq,x D , f for n = 2, 3,... x q x q x 0 ( ) = ( ) with the convention that Dq,xf x f x for the identity operator. { } For a sequence of indeterminates xk k0,wedefine

ω0(x) = 1, n −1 ωn(x) = (x − xk) for n = 1, 2,.... k=0 This paper is organized as follows: in Section 2, we present the first version of divided difference form of the Lagrange–Bürmann formula, in which all the coefficients are in the form of a determinant. { }n In Section 3, generalized Leibniz functional matrices with respect to a sequence xi i=0 are obtained by using the useful Steffensen formula and the properties of divided difference. When all the elements xi in the sequence are repeated, namely x0 = x1 =···= xn = x, we recover all the results in [18]. By means of the factorizations of the generalized Leibniz functional matrices with respect to a sequence which are obtained in Section 3, we establish the second version of divided difference form of the Lagrange–Bürmann formula in Section 4. The coefficients of the second version of divided difference form of the Lagrange–Bürmann formula are explicitly represented by a finite sum. In this section, we also obtain some q-analogues of the Lagrange–Bürmann formula as special cases. In the last section, we obtain the associated matrix inversion.

2. The ﬁrst version of divided difference form of the Lagrange–Bürmann formula

Let F(x) and φ(x) be analytic around x = 0 and φ(0) = 0. Recall that the classical Lagrange– Bürmann expansion formula [17] is a landmark discovery in the history of analysis. Then it is generalized as follows:

∞ n ( ) = ( ) + x , F x F 0 an φ( ) n=1 x P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630 621 where n−1 1 d a = F (x)φn(x) ; n ! n−1 n dx x=0 or

( ) ∞ n F x = x , − φ( )/φ( ) bn φ( ) 1 x x x n=0 x where 1 dn b = F(x)φn(x) . n ! n n dx x=0 In this section, by using the properties of divided differences, we can obtain the ﬁrst version of divided difference form of the Lagrange–Bürmann expansion formula. = k φ , = , ,..., ˆ = / ε =[ , ,..., ]ˆ > Let k i=0 i k 0 1 n, and n,k n k.If n,k xk xk+1 xn n,k for n k and εk,k = 1, then we have the following theorem: φ ( )( = , ,...,∞) ( ) { }∞ Theorem 2.1. Let all i x i 0 1 and h x be analytic functions. For a sequence xi i=0, there must hold n ω ( ) ( ) = ( , )ω ( ) 1 , k x h x A n k n x φ ( ) (1) nk i=0 i x where ··· [ ]( ) 10 0 xk h k ε + , 1 ··· 0 [x , x + ](h + ) k 1 k k k 1 k 1 . . . . A(n, k) = . . . . (2) ε − , ε − , + ··· 1 [x , x + ,...,x − ](h − ) n 1 k n 1 k 1 k k 1 n 1 n 1 εn,k εn,k+1 ··· εn,n−1 [xk, xk+1,...,xn](hn)

Proof. For every m = k, k + 1,...,n, multiplying by m and taking the divided differences at the points x0, x1,...,xm on both sides of (1) respectively, we have m ˆ [x0, x1,...,xm](ωkhm) = A(i, k)[xi, xi+1,...,xm]m,i, i=k which is equivalent to m ˆ [xk, x1,...,xm](hm) = A(i, k)[xi, xi+1,...,xm]m,i i=k by using Lemma 1.1 and the basic facts ⎧ ⎪ ⎨ 1, i = j, [x , x ,...,x ]ω = (3) 0 1 i j ⎩⎪ 0 i = j.

The above is a system of linear equations with respect to A(i, k)(i = k, k + 1,...,n). Therefore, using Cramer’s rule we easily obtain (2). 622 P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630

If we take φi(x) = 1 − yix (i = 0, 1, 2,...)and k = 0in(1), then we reproduce the following result due to Chu [9].

Corollary 2.2. If h(x) is a formal series in x, then we have ∞ (1 − xnyn)ωn(x) h(x) = [x , x ,...,x ]g , n ( − ) 0 1 n n n=0 i=0 1 yix where n −1 gn(x) = h(x) (1 − yix). i=0

n Proof. By the deﬁnition of divided differences, we can see εn,k = (−yi). Therefore i=k+1 ··· [ ] 10 0 xk gk+1 −y + 1 ··· 0 [x , x + ]g + k 1 k k 1 k 2 . . . . . . . . ( , ) = . . . . . A n k (4) − − n1 n1 (−y ) (−y ) ··· 1 [x ,...,x − ]g i i k n 1 n i=k+1 i=k+2 n n (−yi) (−yi) ··· −yn [xk,...,xn]gn+1 i=k+1 i=k+2

Multiplying the ith row by yk+i (i = n − 1, n − 2,...,1) and using the Steffensen formula and the basic properties of divided differences, the determinant (4) takes the following form. ··· [ ] 10 0 xk gk+1 ··· ( − )[ , ] 01 0 xk+1 1 yk+1 xk xk+1 gk+1 . . . . A(n, k) = . . . . ··· ( − )[ ,..., ] 00 1 xn−1 1 yn−1 xk xn−1 gn−1 00··· 0 xn(1 − yn)[xk,...,xn]gn

= xn(1 − yn)[xk,...,xn]gn. If k = 0, then the desired result follows from Theorem 2.1.

3. Generalized Leibniz functional matrices with respect to a sequence

In order to obtain the second kind of divided difference form of the Lagrange–Bürmann formula in Section 4, in this section we give some new definitions and obtain the factorizations of the generalized { }n Leibniz functional matrices with respect to a sequence xi i=0 which extend all of the results in [18]. { }n ( ), ( ),..., ( ) Definition 3.1. Let xi i=0 be a sequence and f0 x f1 x fn x be functions of x such that the nth order derivative exists. The generalized Leibniz functional matrix with varying rows with respect to { }n L R [ , ,..., ] ( + ) × ( + ) xi i=0,denotedby ,{ }n f0 f1 fn ,isan n 1 n 1 matrix, which is defined by n xi i=0 P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630 623 R L n [f , f ,...,f ] n,{xi} = 0 1 n ⎧ i 0 ij ⎨ [ , ,..., ] , = xj xj+1 xi fi if i j , = , , ,..., . ⎩ i j 0 1 2 n (5) 0otherwise,

{ }n ( ), ( ),..., ( ) Deﬁnition 3.2. Let xi i=0 be a sequence and g0 x g1 x gn x be functions of x such that the nth order derivative exists. The generalized Leibniz functional matrix with varying columns with respect { }n L C [ , ,..., ] ( + ) × ( + ) to xi i=0,denotedby ,{ }n g0 g1 gn ,isan n 1 n 1 matrix, which is deﬁned by n xi i=0 C L n [g , g ,...,g ] n,{xi} = 0 1 n ⎧ i 0 ij ⎨ [ , ,..., ] , = xj xj+1 xi gj if i j , = , , ,..., . ⎩ i j 0 1 2 n (6) 0otherwise,

Definition 3.3. With the same assumptions in Definitions 3.1 and 3.2, the generalized Leibniz func- { }n L V [ , ,..., ; tional matrix with varying rows and columns with respect to xi i=0,denotedby ,{ }n f0 f1 fn n xi i=0 g0, g1,...,gn],isan(n + 1) × (n + 1) matrix, which is defined by V L n [f , f ,...,f ; g , g ,...,g ] n,{xi} = 0 1 n 0 1 n ⎧ i 0 ij ⎨ [ , ,..., ]( ) , = xj xj+1 xi figj if i j , = , , ,..., . ⎩ i j 0 1 2 n (7) 0otherwise,

It is easy to see that if fi = gj = f for i, j = 0, 1, 2,...,n,then

R C L ,{ }n [f0, f1,...,fn]=L ,{ }n [g0, g1,...,gn]. n xi i=0 n xi i=0

Therefore, we denote them by L ,{ }n [f ]. n xi i=0 We can explore some algebraic properties for the generalized Leibniz functional matrices with { }n respect to xi i=0.

Theorem 3.1. With the same assumptions in Deﬁnitions 3.1 and 3.2, we can derive

R C L ,{ }n [f0, f1,...,fn]L ,{ }n [g0, g1,...,gn] n xi i=0 n xi i=0 V = L ,{ }n [f0, f1,...,fn; g0, g1,...,gn]. (8) n xi i=0 L R [ , ,..., ]L C [ , ,..., ] = < Proof. It is obvious that ,{ }n f0 f1 fn ,{ }n g0 g1 gn 0fori j.Fori j, n xi i=0 n xi i=0 ij the entry in the ith row and jth column of this matrix is R C L ,{ }n [f0, f1,...,fn]L ,{ }n [g0, g1,...,gn] n xi i=0 n xi i=0 ij i = [xk, xk+1,...,xi]fi[xj, xj+1,...,xk]gj =[xj, xj+1,...,xi](figj). k=j The last equality holds by Lemma 1.1, and this completes the proof of the theorem.

The following corollaries follow easily from Theorem 3.1. 624 P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630

Corollary 3.2 L [ ]L C [ , ,..., ]=L C [ , ,..., ]. n,{x }n f ,{ }n g0 g1 gn ,{ }n fg0 fg1 fgn i i=0 n xi i=0 n xi i=0

Corollary 3.3 L R [ , ,..., ]L [ ]=L R [ , ,..., ]. ,{ }n f0 f1 fn n,{x }n g ,{ }n f0g f1g fng n xi i=0 i i=0 n xi i=0

Corollary 3.4

L ,{ }n [f ]L ,{ }n [g]=L ,{ }n [fg]. n xi i=0 n xi i=0 n xi i=0

Corollary 3.5. With the assumption that f (xi) = 0 (i = 0, 1, 2,...),wehave −1 1 L n [f ]=L ,{ }n . n,{x } n xi = i i=0 i 0 f

L C [ , ,..., ] L R [ , ,..., ] In order to obtain the factorization of ,{ }n g0 g1 gn and ,{ }n f0 f1 fn ,weneed n xi i=0 n xi i=0 the following deﬁnition.

Definition 3.4. With the same assumptions in Definitions 3.1 and 3.2 we define ⎡ ⎤

⎢ In−k 0 ⎥ L C [ , ,..., ]= , , ,{ }k g0 g1 gn ⎣ ⎦ n k xi i=0 L C [ , ,..., ] 0 ,{ }k g0 g1 gk k xi i=0 ⎡ ⎤ L R [f , f ,...,f ] 0 R ⎢ ,{ }k 0 1 k ⎥ L [ , ,..., ]=⎣ k xi i=0 ⎦ , , ,{ }k f0 f1 fn n k xi i=0 0 In−k where In−k is the unit matrix of the size n − k.

If gj = g and fj = f ,forj = 0, 1,...,k,weusethenotation

L [ ]=L C [ , ,..., ], n,k,{x }k g , ,{ }k g g g i i=0 n k xi i=0

L [ ]=LR [ , ,..., ]. n,k,{x }k f , ,{ }k f f f i i=0 n k xi i=0 L C [ , ,..., ] Now we give the result of factorization of ,{ }n g0 g1 gn and n xi i=0 L R [ , ,..., ] ,{ }n f0 f1 fn . n xi i=0

Theorem 3.6. With the same assumptions in Deﬁnition 3.1,wehave

C L ,{ }n [g0, g1,...,gn] n xi i=0 g1 g2 = L , ,{ }n [g ]L , − ,{ }n L , − ,{ }n ... (9) n n xi i=0 0 n n 1 xi i=1 n n 2 xi i=2 g0 g1 gn−1 gn L , ,{ }n L , ,{ } . n 1 xi i=n−1 n 0 xn gn−2 gn−1

Proof. By Corollary 3.2 we have P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630 625

C L ,{ }n [g0, g1,...,gn] (10) n xi i=0 C g1 gn = L ,{ }n [g ]L n 1, ,..., n xi i=0 0 n,{xi} = i 0 g0 g0 ⎡ ⎤ 10 = L [ ] ⎣ ⎦ . n,n,{x }n g0 i i=0 L C [ g1 , g2 ,..., gn ] 0 − ,{ }n n 1 xi i=1 g0 g0 g0 L C g1 , g2 ,..., gn Applying Corollary 3.2 to − ,{ }n in (10)yields n 1 xi i=1 g0 g0 g0

C L ,{ }n [g0, g1,...,gn] n xi i=0 ⎡ ⎤ ⎣ 10⎦ = L , ,{ }n [g ] n n xi i=0 0 g g g L n [ 1 ]L C [ , 2 ,..., n ] 0 n−1,{xi} = − ,{ }n 1 i 1 g0 n 1 xi i=1 g1 g1 ⎡ ⎤ g I 0 = L [ ]L 1 ⎣ 2 ⎦ . n,n,{x }n g0 n,n−1,{x }n i i=0 i i=1 L C [ g2 ,..., gn ] g0 0 − ,{ }n n 2 xi i=2 g1 g1 By the inductive idea and continuing the same procedure, we will prove the theorem.

In a similar manner, we can obtain the following theorem.

Theorem 3.7. With the same assumptions in Deﬁnition 3.2,wehave

R L ,{ }n [f0, f1,...,fn] n xi i=0 f0 f1 f2 = L , ,{ } L 1 L 2 ... (11) n 0 x0 n,1,{xi} = n,2,{xi} = f1 i 0 f2 i 0 f3 fn−1 L n−1 L , ,{ }n [fn] . n,n−1,{xi} = n n xi i=0 i 0 fn

− 1 1 Using Theorem 3.6 and noting L , ,{ }k [g] = L , ,{ }k we can obtain the factorization n k xi i=0 n k xi i=0 g L C [ , ,..., ] of inverse matrix of ,{ }n g0 g1 gn as follows: n xi i=0

Theorem 3.8. With the assumptions that gi(xj) = 0 (i = 0, 1, 2,..., j = i, i + 1,...),theinverse L C [ , ,..., ] matrix of ,{ }n g0 g1 gn has the following factorization: n xi i=0 − C 1 L ,{ }n [g0, g1,...,gn] n xi i=0 gn−1 gn−2 = L , ,{ } L , ,{ }n ... (12) n 0 xn n 1 xi i=n−1 gn gn−1 g1 g0 1 L , − ,{ }n L , − ,{ }n L , ,{ }n . n n 2 xi i=2 n n 1 xi i=1 n n xi i=0 g2 g1 g0 626 P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630

− 1 1 Using Theorem 3.7 and noting L , ,{ }k [f ] = L , ,{ }k we can obtain the factorization n k xi i=0 n k xi i=0 f L R , ,..., of inverse matrix of ,{ }n [f0 f1 fn] as follows: n xi i=0

Theorem 3.9. With the assumptions that fi(xj) = 0 (i = 0, 1, 2,..., j = 0, 1,...,i),theinverse L R [ , ,..., ] matrix of ,{ }n f0 f1 fn has the following factorization: n xi i=0 − R 1 L ,{ }n [f0, f1,...,fn] n xi i=0 1 fn = L , ,{ }n L n−1 ... (13) n n xi i=0 n,n−1,{xi} = fn i 0 fn−1 f3 f2 f1 L 2 L 1 L , ,{ } . n,2,{xi} = n,1,{xi} = n 0 x0 i 0 f2 i 0 f1 f0

4. The second version of divided difference form of the Lagrange–Bürmann formula

In this section, we will establish the second version of divided difference form of the Lagrange– Bürmann formula by means of the factorizations of the generalized Leibniz functional matrices with respect to a sequence which are obtained in Section 3. Now let us introduce the following deﬁnitions.

Deﬁnition 4.1. Let all φi(x)(i = 0, 1,...,n) be functions of x such that the nth derivative exists. For = { , } { }n integers m k 0, N min m n and a sequence xi i=0 the ﬁnite sum N [ , ..., ]φ xji xji+1 xji+1 i (14) i ji i=0 ( = ) ··· (= ) j0 k j1 j2 jN m jN+1 is denoted with emphasis on the order of φi(x) by

D , ,{ }m φ ,φ ,...,φ . m k xi i=0 0 1 n

Deﬁnition 4.2. Let all φi(x)(i = 0, 1,...,n) and h(x) be functions of x such that the nth derivative exists and D , ,{ }m · be given as above. The ﬁnite sum m k xi i=0 m D , ,{ }m φ ,φ ,...,φ [x , x + ,...,x ]h m i xi i=0 0 1 n k k 1 i i=k is denoted by

D , ,{ }m φ ,φ ,...,φ |h. m k xi i=0 0 1 n

Now we are ready to establish the second kind of divided difference form of the Lagrange–Bürmann expansion formula. With the help of Deﬁnitions 4.1 and 4.2, we conclude that φ ( )( = , ,...,∞) ( ) { }∞ Theorem 4.1. Let all i x i 0 1 and h x be analytic functions. For a sequence xi i=0 such that φi(xj) = 0 (i = 0, 1, 2,..., j = i, i + 1,...), there must hold n ω ( ) ( ) = ( , )ω ( ) 1 , k x h x A n k n x φ ( ) (15) nk i=0 i x where P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630 627

A(n, k) = D , ,{ }n φ ,φ ,...,φ |h. (16) n k xi i=0 0 1 n

Proof. Taking the divided differences at both side of (15)atthepointsx0, x1,...,xm (m k) and using Lemma 1.1 and (3), we thereby obtain m n [ , ,..., ] = ( , )[ , ,..., ] 1 xk xk+1 xm h A n k xn xn+1 xm φ (17) n=k i=0 i It is easy to recognize that (17) is equivalent to a multiplication of the lower-triangular matrix of order m + 1 as the following C 1 1 1 L m , ,..., × (A(i, j))( + )×( + ) m,{xi} = m 1 m 1 i 0 φ0 φ0φ1 φ0φ1 ...φm = L ,{ }m [h]. (18) m xi i=0 By virtue of Theorems 3.6 and 3.8 we solve ( ( , )) A i j (m+1)×(m+1) −1 C 1 1 1 = L m , ,..., L ,{ }m [h] m,{xi} = m xi i=0 i 0 φ0 φ0φ1 φ0φ1 ...φm −1 1 1 1 = L , ,{ }m L , − ,{ }m ...L , ,{ } L ,{ }m [h] m m xi i=0 m m 1 xi i=1 m 0 xm m xi i=0 φ0 φ1 φm = L , ,{ } [φ ] L , ,{ }m [φ − ] ...L , ,{ }m [φ ] L ,{ }m [h]. (19) m 0 xm m m 1 xi i=m−1 m 1 m m xi i=0 0 m xi i=0 It is easy to check that the (m, k) entry of the product of the ﬁrst m + 1 matrices on the right side of the last equality turns out to be the form L φ L φ ...L φ m,0,{x } [ m] m,1,{x }m [ m−1] m,m,{x }m [ 0] m i i=m−1 i i=0 m,k = D , ,{ }m <φ ,φ ,...,φ >. m k xi i=0 0 1 m Finally, by equating the (m, k) entryofbothsidesof(19) we therefore have ﬁnished the proof of the theorem.

i If we take xi = q a and k = 0inTheorem4.1, we then have the following corollary.

Corollary 4.2. Let all φi(x)(i = 0, 1,...,∞) and h(x) be analytic functions. We have ∞ n cn (a/x; q) x h(x) = n , n φ ( ) n=0 i=0 i x where k j + −j n D , h(x) n D i, 1 i φ (qji x) q x −ji(ji+1−ji) q x i cn = q (q; q) (q; q) + − k=0 k i ji i=0 ji 1 ji = ··· = k j0 j1 j2 jn jn+1 n x=a

i Let φi(x) = 1 − q x in Corollary 4.2. With the same assumptions in Corollary 4.2, we reproduce the result of Liu [12].

Corollary 4.3. If h(x) is a formal series in x, then we have 628 P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630

∞ ( / ; ) n ( ) = cn a x q n x , h x ( ; ) n=0 x q n+1 where − 2n 1 aq n cn = D , {h(x)(x; q)n} . q x x=a (q; q)n

Proof. It is a direct consequence of Theorem 4.1 by using −1 C 1 1 1 L n , ,..., n,{xi} = n i 0 1 − y x (1 − y x)(1 − y x) = (1 − y x) ⎡ 0 0 1 i 0 i ⎤ n −1 R ⎣ ⎦ = L ,{ }n 1 − x0y0,(1 − x1y1)(1 − y0x),...,(1 − xnyn) (1 − yix) , (20) n xi i=0 i=0 which results from Theorem 3.1 and the fact that

[xk, xk+1,...,xn]n,k = δn,k, where n−1 (1 − xnyn) = (1 − yix) , (x) = i 1 n k k ( − ) i=0 1 yix and the "Kronecker delta" is deﬁned by δn,k = 1ifn = k, and δn,k = 0ifn = k. i i Letting xi = q a, yi = q in (20) we obtain that −1 1 1 1 L C , ,..., n,{qia}n i=0 (x; q) (x; q) (x; q) + 1 2 n 1 = L R − ,( − 2 )( ; ) ,...,( − 2n )( ; ) . ,{ i }n 1 a 1 q a x q 1 1 q a x q n n q a i=0 The desired result received by noticing that n D , {h(x)(x; q)n} [ , ,..., n ] { ( )( ; ) } = q x . a qa q a h x x q n ( ; ) q q n x=a

5. Associated matrix inversion

In this section, some new matrix inversions are presented.

Theorem 5.1. Under the notation and assumption as in Theorem 4.1,wehave ⎛ ⎞ −1 k ⎝ 1 1 1 ⎠ D , ,{ }n , ,..., | ψ n k xi i=0 i φ0 φ1 φn = i 0 nk0 ⎛ ⎞ k ⎝ 1 1 1 ⎠ = D , ,{ }n , ,..., | φ n k xi i=0 i ψ0 ψ1 ψn = i 0 nk0

Proof. It is easily to see that the system of linear equations P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630 629

k n ωk(x) ψi(x) = A(n, k)ωn(x) φi(x) i=0 nk i=0 is equivalent to

k n ωk(x) φi(x) = B(n, k)ωn(x) ψi(x), i=0 nk i=0 where the matrix (B(n, k)) is the inverse of (A(n, k)). Upon applying Theorem 4.1 to these two relations, we immediately assert the theorem.

1 1 The special case of Theorem 5.1 with φi(x) = , ψi(x) = (i = 0, 1, 2,...)can be given 1−yix 1−zix in the following form.

Corollary 5.2. We have the following matrix inversion: [ , ,..., ] −1 = ( − )[ , ,..., ] , xk xk+1 xn fn,k nk0 1 xnzn xk xk+1 xn gn,k nk0 where n−1 = (1 − yix) f , (x) = (1 − x y ) i 0 , n k n n k = (1 − zix) i 0 n−1 = (1 − zix) g , (x) = (1 − x z ) i 0 . n k n n k ( − ) i=0 1 yix

Proof. It is a direct consequence of Theorem 4.1 by using (20). To show the result, we see

(A(i, j))(n+1)×(n+1) −1 C 1 1 1 = L n , ,..., n,{xi} = n i 0 1 − y x (1 − y x)(1 − y x) = (1 − y x) 0 0 1 i 0 i C 1 1 1 × L n , ,..., n,{xi} = n i 0 1 − z x (1 − z x)(1 − z x) = (1 − z x) 0 0 1 i 0 i = [ , ,..., ] . xj xj+1 xi fi,j ij0 Bythesameway,wecanderive ( ( , )) = [ , ,..., ] . B i j (n+1)×(n+1) xj xj+1 xi gi,j ij0 Therefore the result derives easily.

Furthermore, as immediate consequence of Corollary 5.2,weobtain

Corollary 5.3. For arbitrary complex numbers a = 0, b = 0, x, there hold

⎛⎡ ⎤ ⎞−1 ( − 2n) n−k( / ; ) ( n ; ) ⎝⎣ n ⎦ 1 axq b a b q n−k aq x q k ⎠ (bqkx; q) + k n 1 ⎛⎡ ⎤ n ⎞k 0 ( − 2n) n−k( / ; ) ( n ; ) = ⎝⎣ n ⎦ 1 bxq a b a q n−k bq x q k ⎠ k k (aq x; q)n+1 nk0 630 P. Tang, A. Xu / Linear Algebra and its Applications 436 (2012) 618–630

Proof. By noticing that − ( − ) q k n k [ k , k+1 ,..., n ] = n−k ( k ) q x q x q x f Dq,x f q x (q; q)n−k and (λx; q) + μm(q; q) + (λ/μ; q) (λx; q) + m m n = m n m m n , Dq,x (μx; q)n+1 (q; q)n(λx; q)m(μx; q)m+n+1 the result is a direct consequence of Corollary 5.2 under the simultaneous specialization that

i i yi = aq and zi = bq i = 0, 1, 2 ....

Acknowledgments

The authors thank the anonymous referees for their suggestions and comments which improved the quality of this paper. This work was supported by Scientiﬁc Research Foundation of Hangzhou Normal University (No. HSKQ0041), the Zhejiang Province Natural Science Foundation (No. Y6110310), and the Ningbo Natural Science Foundation (No. 2010A610099).

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