Selçuk J. Appl. Math. Selçuk Journal of Vol. 8. No. 1. pp. 9-14 , 2007 Applied Mathematics
On the norms of Hankel matrices with the Motzkin numbers Aynur Yalçıner and Necati Ta¸skara
Department of Mathematics, Selcuk University, 42031 (Campus) Konya, Turkey e-mail:ayalciner@ selcuk.edu.tr, ntaskara@ selcuk.edu.tr
Received: November 11, 2006
Summary:We give bounds for the norm of the Hadamard inverse of Hankel
matrices with Motzkin numbers.Furthermore, we determine an upper bound for the Hadamard product of these matrices.
Key words: Motzkin number; Hadamard inverse; Hankel matrix; norm;
Euclidean norm. 1. Introduction
The Motzkin numbers are defined by the recursion
= −1+ X−2 =0
−2− 0= 1
with generating functions
() =³1 − −p1 − 2 − 32´22
The first few numbers are thus
0 1 2 3 4 5 6 7
1 1 2 4 9 21 51 127
It was shown in [1] that, for the Motzkin numbers, (i) −1 ≤ +1 ( ≥ 1) (ii) −1 3 ( ≥ 1) (iii) lim →∞ −1 = 3.
(1.1) = ⎡ ⎢ ⎢ ⎢ ⎣ 0 1 · · · 1 2 · · · +1 .. . +1 · · · 2 ⎤ ⎥ ⎥ ⎥ ⎦ and (1.2) = ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 · · · 2 3 · · · +1 .. . +1 · · · 2−1 ⎤ ⎥ ⎥ ⎥ ⎦
and obtained following proposition: Proposition (i)det = 1 for all ,
(ii) det = 1, 0, −1 for ≡ 0, 1 (mod 6), ≡ 2, 5 (mod 6) and ≡ 3, 4
(mod 6) respectively [1].
In this study, we have obtained an upper bound for the norm of the Hadamard
inverse of Hankel matrix in the following form:
(1.3) = ⎡ ⎢ ⎢ ⎢ ⎣ 2 3 · · · +1 3 4 · · · +2 .. . +1 +2 · · · 2 ⎤ ⎥ ⎥ ⎥ ⎦
and we have found an upper bound for the Euclidean norm of Hadamard product of these Hankel matrices.
Now, we start with some preliminaries. Let = () and = () be ×
matrices. The Hadamard product of and is defined by ◦ = ()
The Hadamard inverse of (with 0, 1 ≤ ≤ , 1 ≤ ≤ ) is defined by
◦(−1)=³ 1
´ [6].
Let be any × matrix. The norm of the matrix is
kk= ⎛ ⎝ X =1 X =1 || ⎞ ⎠ 1 (1 ≤ ∞) If = ∞ then kk∞= lim →∞kk= max ||
kk = ⎛ ⎝ X =1 X =1 ||2 ⎞ ⎠ 12
and also the spectral norm is kk2=
r max
1≤≤
()
where is × and is the conjugate transpose of the matrix . A function is called a Riemann-zeta function if
() = P∞
=1
1
( 1)
2. Main Results
Theorem 1Let the matrix be as in (1.3). Then
° ° °◦(−1) ° ° ° ≤ s () + 1 4 where 1 ∞.
Proof.From the definition of norm, we have
° ° °◦(−1) ° ° ° = X =1 +1 + X−1 =1 − ++1 For the first term, we have
(2.1) X =1 +1 X =1 1
Let us prove (2.1) by induction. For = 1 inequality (2.1) is true. We shall assume the result is true for − 1 ∈ N. Since ∈ N and +1≥ 2 we have
+1
1
Hence for ∈ N inequality (2.1) is true. We consider the limit case as → ∞ in (2.1) P =1 +1 () On the other hand, for ≥ 1, we write
(2.2) +1 ≥ 2
We can prove (2.2) by induction. For = 1 inequality (2.2) is true. We shall assume that for − 1 ∈ N,
−1 ≥ 2 is true. Since −1 ≤ +1 we have +1 ≥ 2
Consequently for ∈ N inequality (2.2) is true. It is easy to see that for ≥ 4
X−1 =1 − ++1 ≤ 1 4
Using inequality (2.2), we have ++1 4+−3 and therefore, we write
−1 X =1 − ++1 1 4 X−1 =1 − + −3 = 1 4 µ − 1 −2 + − 2 −1 + + 1 2 −4 ¶ Since −1 −2 −2 −1 1 2−4 we obtain −1 X =1 − ++1 1 4 ( − 1)2 −2 For ≥ 5, since −2 ≥ ( − 1)2and 0
(−1)2 −2 1 we have (2.3) X−1 =1 − ++1 ≤ 1 4 Thus the proof is completed.
Corollary 1Let the matrix be as (1.1). Then
° ° °◦(−1) ° ° °≤ s 3 + () + 1 2 where 1 ∞.
Corollary 2Let the matrix be as (1.2). Then
° ° °◦(−1) ° ° °≤ s 1 + () + 1 3
where 1 ∞ .
Theorem 2Let and matrices as in (1.3) and (1.2), respectively.Then
° ° °◦ ◦(−1) ° ° ° ≤ 3
where k·k is the Euclidean norm.
Proof.From the definition of Euclidean norm, we have ° ° °◦ ◦(−1) ° ° °2 = X =1 µ +1 ¶2 + −1 X =1 ( − ) µ ++1 + ¶2 since +1 3 ( ≥ 1), we obtain ° ° °◦ ◦(−1) ° ° °2 = 9 X =1 + 9 X−1 =1 ( − ) = 92 Hence, ° ° °◦ ◦(−1) ° ° ° ≤ 3
Example 1Let = 2. For inequality (2.1) , we have the following values: n P =1 +1 P =1 1 2 3 4 5 10 20 30 40 50 100 0.25000000 0.37500000 0.41203704 0.42110735 0.42348212 0.42348248 0.42348248 0.42348248 0.42348248 0.42348248 1.00000000 1.25000000 1.36111116 1.42361116 1.53976774 1.59366302 1.61103916 1.61961925 1.62473297 1.63488400
n P− =1 − ++1 1 4 2 3 4 5 6 7 8 9 10 11 12 0.01234568 0.00491962 0.00128699 0.00027983 0.00005435 0.00000978 0.00000166 0.00000027 0.00000004 0.00000001 0.00000000 0.01234568 0.01234568 0.01234568 0.01234568 0.01234568 0.01234568 0.01234568 0.01234568 0.01234568 0.01234568 0.01234568 References
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3. R. Donaghey, L.W. Shapiro, Motzkin numbers, Journal of Combinatorial Theory, Series A 23 (1977), 291-301.
4.R.Stanley, Enumerative Combinatorics, Vol II, Cambridge Univ. Press, 1999. 5.R.A.Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
6. R.A.Horn, The Hadamard product, in: C.R. Johnson (Ed.), Proc. Appl. Math., vol. 40. American Mathematical Society, Providence, Rhode Island. (1990) 87-169.
