The Explicit Identities for Spectral Norms of Circulant-Type Matrices Involving Binomial Coefficients and Harmonic Numbers

The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers [1]. In numerical analysis, circulant matrices (named “premultipliers” in numerical methods) are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. Furthermore, circulant, skew-circulant, and g-circulant matrices play important roles in various applications, such as image processing, coding, and engineering model. For more details, please refer to [2–13] and the references therein. The skew-circulant matrices were collected to construct preconditioners for LMF-based ODE codes; Hermitian and skew-Hermitian Toeplitz systems were considered in [14– 17]; Lyness employed a skew-circulant matrix to construct sdimensional lattice rules in [18]. Recently, there are lots of research on the spectral distribution and norms of circulanttype matrices. In [19], the authors pointed out the processes based on the eigenvalue of circulant-type matrices and the convergence to a Poisson random measure in vague topology. There were discussions about the convergence in probability and distribution of the spectral normof circulanttype matrices in [20]. The authors in [21] listed the limiting spectral distribution for a class of circulant-type matrices with heavy tailed input sequence. Ngondiep et al. showed that the singular values of g-circulants in [22]. Solak established the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries in [23]. İpek investigated an improved estimation for spectral norms in [24]. In this paper, we derive some explicit identities of spectral norms for some circulant-type matrices with product of binomial coefficients with harmonic numbers. The outline of the paper is as follows. In Section 2, the definitions and preliminary results are listed. In Section 3, the spectral norms of some circulant matrices are studied. In Section 4, the formulae of spectral norms for skew-circulant matrices are established. Section 5 is devoted to investigate the explicit formulae for g-circulant matrices. The numerical tests are given in Section 6.


Introduction
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers [1]. In numerical analysis, circulant matrices (named "premultipliers" in numerical methods) are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. Furthermore, circulant, skew-circulant, and -circulant matrices play important roles in various applications, such as image processing, coding, and engineering model. For more details, please refer to [2][3][4][5][6][7][8][9][10][11][12][13] and the references therein. The skew-circulant matrices were collected to construct preconditioners for LMF-based ODE codes; Hermitian and skew-Hermitian Toeplitz systems were considered in [14][15][16][17]; Lyness employed a skew-circulant matrix to constructdimensional lattice rules in [18]. Recently, there are lots of research on the spectral distribution and norms of circulanttype matrices. In [19], the authors pointed out the processes based on the eigenvalue of circulant-type matrices and the convergence to a Poisson random measure in vague topology. There were discussions about the convergence in probability and distribution of the spectral norm of circulanttype matrices in [20]. The authors in [21] listed the limiting spectral distribution for a class of circulant-type matrices with heavy tailed input sequence. Ngondiep et al. showed that the singular values of -circulants in [22]. Solak established the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries in [23].İpek investigated an improved estimation for spectral norms in [24].
In this paper, we derive some explicit identities of spectral norms for some circulant-type matrices with product of binomial coefficients with harmonic numbers.
The outline of the paper is as follows. In Section 2, the definitions and preliminary results are listed. In Section 3, the spectral norms of some circulant matrices are studied. In Section 4, the formulae of spectral norms for skew-circulant matrices are established. Section 5 is devoted to investigate the explicit formulae for -circulant matrices. The numerical tests are given in Section 6.

Preliminaries
The binomial coefficients are defined by ( ) for all natural numbers at once by Note that ( ) is the th binomial coefficient of . It is clear that ( 0 ) = 1, ( ) = 1, and ( ) = 0, for > .

Mathematical Problems in Engineering
The generalized harmonic numbers are defined to be partial sums of the harmonic series [1]: For = 0 in particular, they reduce to classical harmonic numbers: We recall the following harmonic number identities [1]: Definition 1 (see [6,8]). A circulant matrix is an × complex matrix with the following form: The first row of is ( 0 , 1 , . . . , −1 ); its ( + 1)th row is obtained by giving its th row a right circular shift by one position.
Definition 3 (see [21,25]). A -circulant matrix is an × complex matrix with the following form: where is a nonnegative integer and each of the subscripts is understood to be reduced modulo .
Mathematical Problems in Engineering 3 The first row of is ( 0 , 1 , . . . , −1 ); its ( + 1)th row is obtained by giving its th row a right circular shift by positions (equivalently, mod positions). Note that = 1 or = + 1 yields the standard circulant matrix. If = − 1, then we obtain the so-called reverse circulant matrix [21].
Definition 4 (see [26]). The spectral norm ‖ ⋅ ‖ 2 of a matrix with complex entries is the square root of the largest eigenvalue of the positive semidefinite matrix * : where * denotes the conjugate transpose of . Therefore if is an × real symmetric matrix or is a normal matrix, then where 1 , 2 , . . . , are the eigenvalues of .

Spectral Norms of Some Circulant Matrices
Now, we will analyse spectral norms of some given circulant matrices, whose entries are binomial coefficients combined with harmonic numbers. Our main results for those matrices are stated as follows.

Mathematical Problems in Engineering
Employing the same approaches, we get the following corollary. 4 be as in (5), and the first row of 4 is

Spectral Norms of Skew-Circulant Matrices
An odd-order alternative skew-circulant matrix is defined as follows, where is even.
Since the skew-circulant matrix is normal, we deduce that If is even, then + 1 is odd. We declare that sc = −1 is an eigenvalue of sc ; then we calculate the corresponding eigenvalue of 5 as follows: where we had employed (14). Noticing (34), we claim that̂( 5 ) is the maximum of | ( 5 )|, which means Thus, from (4) we obtain This completes the proof.
Similarly, we can calculate the identity for 6 .

Spectral Norms of -Circulant Matrices
Inspired by the above propositions, we analyse spectral norms of some given -circulant matrices in this section.

Numerical Examples
Example 1. In this example, we give the numerical results for 1 and 2 . Comparing the data in Table 1, we declare that the identities of spectral norms for ( = 1, 2) hold.
With the help of data in Table 2, it is clear that the identities of spectral norms for ,̃( = 3, 4) hold.
Combining the data in Table 3, we deduce that the identities of spectral norms for ,̃( = 5, 6) hold.

Example 4.
In this example, we show numerical results for 7 and 8 .
Considering the data in Table 4, we deduce that the identities of spectral norms for ( = 7, 8) hold.
The above results demonstrate that the identities of spectral norms for the given matrices hold.

Conclusion
This paper had discussed the explicit formulae for identical estimations of spectral norms for circulant, skew-circulant and -circulant matrices, whose entries are binomial coefficients combined with harmonic numbers. Furthermore, it is easy to take other entries to obtain more interesting identities, and the same approaches can be used to verify those identities. Furthermore, explicit formulas for both Table 1: Spectral norms of ( = 1, 2), 1, = 2( 2 ) 2 ( 2 − ), and 2, = ( 3 ) 2 (2 3 − 2 − ).    norms ‖ ‖ and ‖ −1 ‖ help us to estimate the so-called condition number. It is an interesting problem to investigate the properties of ( = 1, 2, . . . , 8), such as the explicit formulations for determinants and inverses, by just using the entries in the first row.