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

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jianwei Zhou ◽  
Xiangyong Chen ◽  
Zhaolin Jiang
Keyword(s):  
Circulant Matrix ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Numerical Tests ◽  
Skew Circulant ◽  
Explicit Formulae

The explicit formulae of spectral norms for circulant-type matrices are investigated; the matrices are circulant matrix, skew-circulant matrix, andg-circulant matrix, respectively. The entries are products of binomial coefficients with harmonic numbers. Explicit identities for these spectral norms are obtained. Employing these approaches, some numerical tests are listed to verify the results.

SPECTRAL NORMS OF CIRCULANT-TYPE MATRICES WITH BINOMIAL COEFFICIENTS AND HARMONIC NUMBERS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350076 ◽  
Author(s):  
JIANWEI ZHOU ◽  
ZHAOLIN JIANG
Keyword(s):  
Circulant Matrices ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Numerical Tests ◽  
Skew Circulant

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skew-circulant, and g-circulant matrices. The entries are product of Binomial coefficients with Harmonic numbers. We obtain explicit identities for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

scholarly journals The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jin-jiang Yao ◽  
Zhao-lin Jiang
Keyword(s):  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Skew Circulant ◽  
The Relationship

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

scholarly journals Congruences involving alternating sums related to harmonic numbers and binomial coefficients

2020 ◽  
Vol 26 (4) ◽  
pp. 39-51
Author(s):  
Laid Elkhiri ◽  
◽  
Miloud Mihoubi ◽  
Abdellah Derbal ◽  
◽  
...  
Keyword(s):  
Prime Number ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Arithmetic Properties ◽  
Alternating Sums ◽  
Generalized Harmonic Numbers

In 2017, Bing He investigated arithmetic properties to obtain various basic congruences modulo a prime for several alternating sums involving harmonic numbers and binomial coefficients. In this paper we study how we can obtain more congruences modulo a power of a prime number p (super congruences) in the ring of p-integer \mathbb{Z}_{p} involving binomial coefficients and generalized harmonic numbers.

scholarly journals Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhaolin Jiang ◽  
Tingting Xu ◽  
Fuliang Lu
Keyword(s):  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Cyclic Group ◽  
Functional Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Linear Vector ◽  
Skew Circulant ◽  
Linear Vector Space

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra ofn×ncomplex skew-circulant matrices are displayed in this paper.

A Harmonic-Canceling Synthesizer using Skew-Circulant-Matrix-Based Coefficient Generator

2020 ◽  
Author(s):  
Guillermo G. Garayar-Leyva ◽  
Hatem Osman ◽  
Johan J. Estrada-Lopez ◽  
Edgar Sanchez-Sinencio
Keyword(s):  
Circulant Matrix ◽  
Skew Circulant

scholarly journals Supercongruences related to 3F2(1) involving harmonic numbers

2018 ◽  
Vol 14 (04) ◽  
pp. 1093-1109 ◽  
Author(s):  
Roberto Tauraso
Keyword(s):  
Infinite Series ◽  
Binomial Coefficients ◽  
Harmonic Numbers

We provide various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

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