Explicit Inverse of the Pascal Matrix Plus One

EXPLICIT INVERSE OF THE PASCAL MATRIX PLUS ONE

SHENG-LIANG YANG AND ZHONG-KUI LIU

Received 5 June 2005; Revised 21 September 2005; Accepted 5 December 2005

This paper presents a simple approach to invert the matrix Pn + In by applying the Euler polynomials and Bernoulli numbers, where Pn is the Pascal matrix.

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1. 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 [1, 3–5]. For any integer n>0, the n × n Pascal matrix Pn is defined with the binomial coefficients by ⎧⎛ ⎞ ⎪ ⎪ i − 1 ⎨⎝ ⎠ if i ≥ j ≥ 1, P i j = n( , ) ⎪ j − 1 (1.1) ⎪ ⎩0 otherwise.

−1 It is known that the n × n inverse matrix Pn is given by ⎧ ⎛ ⎞ ⎪ ⎪ i − 1 ⎨(−1)i− j ⎝ ⎠ if i ≥ j ≥ 1, P i j = n( , ) ⎪ j − 1 (1.2) ⎪ ⎩0 otherwise.

The Hadamard product A ◦ B of two matrices is the matrix obtained by coordinate- wise multiplication: (A ◦ B)(i, j) = A(i, j)B(i, j). Let Γn be the n × n lower triangular ma- trices defined by ⎧ ⎨(−1)i− j if i ≥ j ≥ 1, Γn(i, j) = (1.3) ⎩0 otherwise,

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 90901, Pages 1–7 DOI 10.1155/IJMMS/2006/90901 2 Explicit inverse of the Pascal matrix plus one

−1 then the inverse of the Pascal matrix can be represented as the Hadamard product Pn = Pn ◦ Γn.Forexample,ifn = 5, then

⎛ ⎞ 10000 ⎜ ⎟ ⎜ ⎟ ⎜11000⎟ ⎜ ⎟ ⎜ ⎟ P = ⎜12100⎟, 5 ⎜ ⎟ ⎜ ⎟ ⎝13310⎠ 14641 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 10000 10000 10000 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜−11000⎟ ⎜11000⎟ ⎜−11 0 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ P 1 = ⎜ 1 −21 00⎟ = ⎜12100⎟ ◦ ⎜ 1 −11 00⎟. 5 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝−13−310⎠ ⎝13310⎠ ⎝−11−110⎠ 1 −46−41 14641 1 −11−11 (1.4)

Now we consider the sum of the Pascal matrix and the identity matrix Pn + In,where In is the n × n identity matrix. We call Pn + In the Pascal matrix plus one simply. An interesting fact is that the inverse of Pn + In is related to Pn closely. For instance,

⎛ ⎞ 20 0 0 00 ⎜ ⎟ ⎜ ⎟ ⎜12 0 0 00⎟ ⎜ ⎟ ⎜ ⎟ ⎜12 2 0 00⎟ P I = ⎜ ⎟ 6 + 6 ⎜ ⎟, ⎜13 3 2 00⎟ ⎜ ⎟ ⎜ ⎟ ⎝14 6 4 20⎠ 1 5 10 10 5 2

⎛ ⎞ 1 ⎜ 00000⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜− 1 1 ⎟ ⎜ 0000⎟ ⎜ 4 2 ⎟ ⎜ ⎟ ⎜ 2 1 ⎟ ⎜ 0 − 000⎟ − ⎜ ⎟ P I 1 = ⎜ 4 2 ⎟ 6 + 6 ⎜ ⎟ ⎜ 1 3 1 ⎟ ⎜ 0 − 00⎟ ⎜ ⎟ ⎜ 8 4 2 ⎟ ⎜ ⎟ ⎜ 4 4 1 ⎟ ⎜ 0 0 − 0⎟ ⎜ 8 4 2 ⎟ ⎝ 1 10 5 1⎠ − 0 0 − 4 8 4 2 S.-L. Yang and Z.-K. Liu 3 ⎛ ⎞ 1 ⎜ 00000⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎜− 1 1 ⎟ ⎜ 0000⎟ 10 0 0 00 ⎜ 4 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜11 0 0 00⎟ ⎜ 1 1 ⎟ ⎜ ⎟ ⎜ − ⎟ ⎜ ⎟ ⎜ 0 000⎟ = ⎜12 1 0 00⎟ ◦ ⎜ 4 2 ⎟. ⎜ ⎟ ⎜ ⎟ ⎜13 3 1 00⎟ ⎜ 1 1 1 ⎟ ⎜ ⎟ ⎜ 0 − 00⎟ ⎝14 6 4 10⎠ ⎜ ⎟ ⎜ 8 4 2 ⎟ ⎜ ⎟ 1 5 10 10 5 1 ⎜ 1 1 1 ⎟ ⎜ 0 0 − 0 ⎟ ⎜ 8 4 2 ⎟ ⎝ 1 1 1 1 ⎠ − 0 0 − 4 8 4 2 (1.5) {a }∞ P This suggests that there may exist a sequence of constants n n=0 such that ( n + − In) 1 = Pn ◦ Δn,wherethematrixΔn is a lower triangular matrix with generic element Δn(i, j) = ai− j when i ≥ j. Aggarwala and Lamoureux [2] have showed that these con- stants are values of the Dirichlet eta function evaluated at negative integers, or more gen- erally, certain polylogarithm functions evaluated at the number −1. In this note, we will give a new simple approach to invert the matrix Pn + In by applying the Euler polyno- mials. As a result, we will show that these constants are values of the Euler polynomials evaluated at the number 0. The Euler polynomials En(x) are defined by means of the following generating func- tion (see [7]): ∞ tn etx E x = 2 n( )n et , (1.6) n=0 ! +1 ∞ E x E x tn/n = ∞ E x tn/n ∞ E x tn/n = since n=0( n( +1)+ n( ))( !) n=0 n( +1)( !) + n=0 n( )( !) et(x+1)/ et etx/ et = etx = ∞ xn tn/n ffi 2 ( +1)+2 ( +1) 2 n=0 2 ( !). Comparing the coe cients of tn/n! in this equation, we obtain

n En(x +1)+En(x) = 2x , n ≥ 0. (1.7)

The following lemmas are well known and can be found in [9], we give a short proof for the sake of completeness. Lemma 1.1. For all n ≥ 0, n n E x y = E x yn−k n( + ) k k( ) , (1.8) k= 0 n n E x = E x . n( +1) k k( ) (1.9) k=0 ∞ E x y tn/n = et(x+y)/ et = etx/ et ety = ∞ E x tn/ Proof. n=0 n( + )( !) 2 ( +1) (2 ( +1)) ( n=0 n( )( n ∞ yn tn/n = ∞ n n E x yn−k tn/n ffi !))( n=0 ( !)) n=0( k=0 k k( ) )( !). Comparing the coe cients of tn/n! in this equation, we obtain (1.8). In particular, when y = 1, we get (1.9). 4 Explicit inverse of the Pascal matrix plus one

From (1.7)and(1.9), we obtain n n 1 E x 1E x = xn n ≥ . k k( )+ n( ) , 0 (1.10) 2 k=0 2 x = E y = n n E yk If we set 0in(1.8), we get n( ) k=0 k n−k(0) , that is, n n E x = E xk n ≥ . n( ) k n−k(0) , 0 (1.11) k=0

T Let E(x)andX(x)bethen × 1matricesdefinedbyE(x) = [E0(x),E1(x),...,En−1(x)] , n− T X(x) = [1,x,...,x 1] ,andletE¯n be n × n lower triangular matrices defined by ⎧⎛ ⎞ ⎪ i − ⎪⎝ 1⎠ ⎨ Ei− j (0) if i ≥ j ≥ 1, j − E¯n(i, j) = 1 (1.12) ⎪ ⎩⎪ 0 otherwise.

Then (1.10), (1.11) can be represented as matrix equations, respectively,

1 Pn + In E(x) = X(x), 2 (1.13) E(x) = E¯nX(x).

Thus, we have

−1 Pn + In 1 = E¯n 2 ⎛ ⎞ 0 ⎜ E ··· ⎟ ⎜ 0(0) 0 0 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 ⎟ ⎜ E (0) E (0) 0 ··· 0 ⎟ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜ 2 E 2 E 2 E ··· ⎟ = ⎜ 2(0) 1(0) 0(0) 0 ⎟. ⎜ 0 1 2 ⎟ 2 ⎜ ⎟ ⎜ ⎟ ⎜ . . . . ⎟ ⎜ ...... ⎟ ⎜ . . . . . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ n − 1 n − 1 n − 1 n − 1 ⎠ En− (0) En− (0) En− (0) ··· E (0) 0 1 1 2 2 3 n − 1 0 (1.14)  S.-L. Yang and Z.-K. Liu 5

The Bernoulli numbers Bn are defined by (see [7])

∞ tn t B = . n n et − (1.15) n=0 ! 1

It is known (see [6, 8]) that the Euler polynomials can be expressed by the Bernoulli num- bers as n+1 n E x = 1 − k+1 +1 B xn+1−k. n( ) n 2 2 k k (1.16) +1k=1

Putting x = 0in(1.16)gives n+1 2 1 − 2 Bn+1 En(0) = , (1.17) n +1 for all integers n ≥ 0. Therefore, we obtain an explicit inverse of the Pascal matrix plus one as follows.

− Theorem 1.2. For n ≥ 1,then × n inverse matrix Qn = (Pn + In) 1 is given by ⎧ ⎛ ⎞ ⎪ i − ⎨⎪ 1 ⎝ 1⎠ Ei− j (0) if i ≥ j ≥ 1, Qn(i, j) = 2 j − 1 (1.18) ⎪ ⎩⎪ 0 if i

− In view of the Hadamard product, the inverse matrix (Pn + In) 1 is the Hadamard product of the Pascal matrix Pn and the matrix Δn,whereΔn is the n × n lower triangular matrices defined by ⎧ ⎨⎪ 1 Ei− j(0) if i ≥ j ≥ 1, Δn(i, j) = 2 (1.20) ⎩⎪ 0ifi

The two functions, Euler(n,x) and Bernoulli(n), in the combinat library of the com- puter algebra system Maple are very useful in obtaining the matrix Qn.Forexample,for n = 8, we get

⎛ ⎞ 1 ⎜ 0 000000⎟ ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 ⎟ ⎜− 000000⎟ ⎜ 4 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 ⎟ ⎜ 0 − 00000⎟ ⎜ 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 3 1 ⎟ ⎜ 0 − 0000⎟ ⎜ 8 4 2 ⎟ − ⎜ ⎟ Q = (P + I ) 1 = ⎜ ⎟. (1.22) 8 8 8 ⎜ 1 1 ⎟ ⎜ 0 0 −1 000⎟ ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ ⎟ ⎜ 1 5 5 1 ⎟ ⎜− 0 0 − 00⎟ ⎜ ⎟ ⎜ 4 4 4 2 ⎟ ⎜ ⎟ ⎜ 3 5 3 1 ⎟ ⎜ 0 − 0 0 − 0⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎝ 17 21 35 7 1⎠ 0 − 0 0 − 16 4 8 4 2

Note that Qn(i, j) = 0wheneveri

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 [1]L.AcetoandD.Trigiante,The matrices of Pascal and other greats, The American Mathematical Monthly 108 (2001), no. 3, 232–245. [2] R. Aggarwala and M. P. Lamoureux, Inverting the Pascal matrix plus one, The American Mathe- matical Monthly 109 (2002), no. 4, 371–377. [3] M. Bayat and H. Teimoori, Pascal K-eliminated functional matrix and its property, Linear Algebra and Its Applications 308 (2000), no. 1–3, 65–75. [4] R. Brawer and M. Pirovino, ThelinearalgebraofthePascalmatrix, Linear Algebra and Its Appli- cations 174 (1992), 13–23. [5] G.S.CallandD.J.Velleman,Pascal’s matrices, The American Mathematical Monthly 100 (1993), no. 4, 372–376. [6] G.-S. Cheon, A note on the Bernoulli and Euler polynomials,AppliedMathematicsLetters16 (2003), no. 3, 365–368. [7] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions,D.Reidel,Dor- drecht, 1974. 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, 2000. [9]H.M.SrivastavaandA.´ Pinter,´ Remarks on some relationships between the Bernoulli and Euler polynomials,AppliedMathematicsLetters17 (2004), no. 4, 375–380.

Sheng-liang Yang: Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China E-mail address: [email protected]

Zhong-kui Liu: Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China E-mail address: [email protected] Mathematical Problems in Engineering

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