EXPLICIT INVERSE OF THE PASCAL MATRIX PLUS ONE

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1 EXPLICIT INVERSE OF THE PASCAL MATRIX PLUS ONE SHENG-LIANG YANG AND ZHONG-KUI LIU Received 5 June 005; Revised September 005; Accepted 5 December 005 This paper presents a simple approach to invert the matrix P n + I n by applying the Euler polynomials and Bernoulli numbers, where P n is the Pascal matrix. Copyright 006. All rights reserved.. Introduction The Pascal matrix has been known since ancient times, and it arises in many different areas of mathematics. However, it has been studied carefully only recently, see [, 3 5]. For any integer n>0, the n n Pascal matrix P n is defined with the binomial coefficients by i if i j, P n (i, j) = j (.) 0 otherwise. It is known that the n n inverse matrix Pn is given by ( ) i j i if i j, P n (i, j) = j 0 otherwise. (.) The Hadamard product A B of two matrices is the matrix obtained by coordinatewise multiplication: (A B)(i, j) = A(i, j)b(i, j). Let Γ n be the n n lower triangular matrices defined by ( ) i j if i j, Γ n (i, j) = 0 otherwise, (.3) International Mathematics and Mathematical Sciences Volume 006, Article ID 9090, Pages 7 DOI 0.55/IJMMS/006/9090

2 Explicit inverse of the Pascal matrix plus one then the inverse of the Pascal matrix can be represented as the Hadamard product P n = P n Γ n.forexample,ifn = 5, then P 5 = 0 0, P5 = = (.4) Now we consider the sum of the Pascal matrix and the identity matrix P n + I n,where I n is the n n identity matrix. We call P n + I n the Pascal matrix plus one simply. An interesting fact is that the inverse of P n + I n is related to P n closely. For instance, P 6 + I 6 =, P6 + I 6 =

3 S.-L. Yang and Z.-K. Liu = (.5) This suggests that there may exist a sequence of constants {a n } n=0 such that (P n + I n ) = P n Δ n,wherethematrixδ n is a lower triangular matrix with generic element Δ n (i, j) = a i j when i j. Aggarwala and Lamoureux [] have showed that these constants are values of the Dirichlet eta function evaluated at negative integers, or more generally, certain polylogarithm functions evaluated at the number. In this note, we will give a new simple approach to invert the matrix P n + I n by applying the Euler polynomials. As a result, we will show that these constants are values of the Euler polynomials evaluated at the number 0. The Euler polynomials E n (x) are defined by means of the following generating function (see [7]): n=0 E n (x) tn n! = etx e t +, (.6) since n=0 (E n (x +)+E n (x))(t n /n!) = n=0 E n (x +)(t n /n!) + n=0 E n (x)(t n /n!) = e t(x+) /(e t +)+e tx /(e t +)= e tx = n=0 x n (t n /n!). Comparing the coefficients of t n /n! in this equation, we obtain E n (x +)+E n (x) = x n, n 0. (.7) The following lemmas are well known and can be found in [9], we give a short proof for the sake of completeness. Lemma.. For all n 0, n n E n (x + y) = E k (x)y n k, (.8) k n n E n (x +)= E k (x). (.9) k Proof. n=0 E n (x + y)(t n /n!) = e t(x+y) /(e t +)= (e tx /(e t +))e ty = ( n=0 E n (x)(t n / n!))( n=0 y n (t n /n!)) = n=0 ( ( n n k) Ek (x)y n k )(t n /n!). Comparing the coefficients of t n /n! in this equation, we obtain (.8). In particular, when y =, we get (.9).

4 4 Explicit inverse of the Pascal matrix plus one From (.7)and(.9), weobtain n n E k (x)+ k E n(x) = x n, n 0. (.0) If we set x = 0in(.8), we get E n (y) = n ( n k) En k (0)y k, that is, E n (x) = n n E n k (0)x k, n 0. (.) k Let E(x)andX(x)bethen matricesdefinedbye(x) = [E 0 (x),e (x),...,e n (x)] T, X(x) = [,x,...,x n ] T,andletĒ n be n n lower triangular matrices defined by i E i j (0) if i j, Ē n (i, j) = j 0 otherwise. Then (.0), (.) can be represented as matrix equations, respectively, (.) Pn + I n E(x) = X(x), E(x) = Ē n X(x). (.3) Thus, we have ( Pn + I n ) = Ēn 0 E 0 0 (0) E 0 (0) E 0 (0) 0 0 = E 0 (0) E (0) E 0 (0) n n n n E 0 n (0) E n (0) E n 3 (0) E n 0 (0) (.4)

5 S.-L. Yang and Z.-K. Liu 5 The Bernoulli numbers B n are defined by (see [7]) n=0 B n t n n! = t e t. (.5) It is known (see [6, 8]) that the Euler polynomials can be expressed by the Bernoulli numbers as Putting x = 0in(.6)gives E n (x) = n+ ( k+ ) n + B k x n+ k. (.6) n + k k= E n (0) = ( n+) B n+, (.7) n + for all integers n 0. Therefore, we obtain an explicit inverse of the Pascal matrix plus one as follows. Theorem.. For n,then n inverse matrix Q n = (P n + I n ) is given by i E i j (0) if i j, Q n (i, j) = j 0 if i<j; (.8) or i ( i j+ ) B i j+ if i j, Q n (i, j) = j i j + 0 if i<j. (.9) In view of the Hadamard product, the inverse matrix (P n + I n ) is the Hadamard product of the Pascal matrix P n and the matrix Δ n,whereδ n is the n n lower triangular matrices defined by or Δ n (i, j) = E i j(0) if i j, 0 ifi<j; ( i j+ ) B i j+ if i j, Δ n (i, j) = i j + 0 ifi<j. (.0) (.)

6 6 Explicit inverse of the Pascal matrix plus one The two functions, Euler(n,x) and Bernoulli(n), in the combinat library of the computer algebra system Maple are very useful in obtaining the matrix Q n.forexample,for n = 8, we get Q 8 = (P 8 + I 8 ) 8 4 =. (.) Note that Q n (i, j) = 0wheneveri<jor i = j +,j +4,j +6,... Acknowledgments This work is supported by Development Program for Outstanding Young Teachers in Lanzhou University of Technology and the NSF of Gansu Province of China. The authors wish to thank the referees for many valuable comments and suggestions that led to the improvement and revision of this note. References [] L. Aceto and D. Trigiante, The matrices of Pascal and other greats, The American Mathematical Monthly 08 (00), no. 3, [] R. Aggarwala and M. P. Lamoureux, Inverting the Pascal matrix plus one, The American Mathematical Monthly 09 (00), no. 4, [3] M. Bayat and H. Teimoori, Pascal K-eliminatedfunctional matrixanditsproperty, Linear Algebra and Its Applications 308 (000), no. 3, [4] R.Brawer and M.Pirovino,ThelinearalgebraofthePascalmatrix, Linear Algebra and Its Applications 74 (99), 3 3. [5] G. S. Call and D. J. Velleman, Pascal s matrices, The American Mathematical Monthly 00 (993), no. 4, [6] G.-S. Cheon, A note on the Bernoulli and Euler polynomials, AppliedMathematicsLetters6 (003), no. 3, [7] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, D. Reidel, Dordrecht, 974.

7 S.-L. Yang and Z.-K. Liu 7 [8] K. H. Rosen, J. G. Michaels, J. L. Gross, J. W. Grossman, and D. R. Shier (eds.), Handbook of Discrete and Combinatorial Mathematics, CRC Press, Florida, 000. [9] H.M.SrivastavaandÁ. Pintér, Remarks on some relationships between the Bernoulli and Euler polynomials, AppliedMathematicsLetters7 (004), no. 4, Sheng-liang Yang: Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu , China address: slyang@lut.cn Zhong-kui Liu: Department of Mathematics, Northwest Normal University, Lanzhou, Gansu , China address: liuzk@nwnu.edu.cn

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