On the Frobenius norms of circulant matrices with Ducci sequences and Narayana and Gaussian Narayana numbers

Roji Bala; Vinod Mishra

Makale detayları

On the Frobenius norms of circulant matrices with Ducci sequences and Narayana and Gaussian Narayana numbers

Yazarlar:

Roji Bala | Vinod Mishra

Yayıncı:

Ankara University

Ana Başlıklar:

Algebra and Number Theory

Dergi:

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Vol.74 Issue 3, pp.460-477

Belge Tipi:

Makale

Yayın Tarihi:

9/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001357871

pISSN:

1303-5991

Anahtar kelimeler:

Özet:

 In the present paper, we obtain identities for Narayana numbers, like the sum of terms with even and odd subscripts, the sum of products of consecutive terms and the sum of squares of terms. Then, we find images DN and D2N of n-tuple NN_1,  N_2, N_3,  ...  , N_n of Narayana numbers under a map D:mathbbC rightarrow mathbbC defined as Dz_1,  z_2,  ...  ,  z_nlvert z_2-z_1rvert,  lvert z_3-z_2rvert,  ...  , lvert z_n-z_n-1rvert,  lvert z_n-z_1rvert . We are then determined the circulant, skew-circulant, and semi-circulant matrices of these images. We have been discovered Frobenius norms of these circulant matrices and relations among these norms. In addition, we find DG and D2G by taking the n-tuple GGN_1, GN_2,  ...  ,  GN_n of Gaussian Narayana numbers. After that, we create circulant, semi-circulant, and skew-circulant matrices of G,  DG,  D2G, determine their Frobenius norms, and derive relationships between them. Then, we obtain relations between norms of matrices of Narayana numbers and Gaussian Narayana numbers. Finally, coding and decoding methods with the use of circulant matrices of Narayana numbers and Gaussian Narayana numbers have been introduced.

 In the present paper, we obtain identities for Narayana numbers, like the sum of terms with even and odd subscripts, the sum of products of consecutive terms and the sum of squares of terms. Then, we find images DN and D2N of n-tuple NN_1,  N_2, N_3,  ...  , N_n of Narayana numbers under a map D:mathbbC rightarrow mathbbC defined as Dz_1,  z_2,  ...  ,  z_nlvert z_2-z_1rvert,  lvert z_3-z_2rvert,  ...  , lvert z_n-z_n-1rvert,  lvert z_n-z_1rvert . We are then determined the circulant, skew-circulant, and semi-circulant matrices of these images. We have been discovered Frobenius norms of these circulant matrices and relations among these norms. In addition, we find DG and D2G by taking the n-tuple GGN_1, GN_2,  ...  ,  GN_n of Gaussian Narayana numbers. After that, we create circulant, semi-circulant, and skew-circulant matrices of G,  DG,  D2G, determine their Frobenius norms, and derive relationships between them. Then, we obtain relations between norms of matrices of Narayana numbers and Gaussian Narayana numbers. Finally, coding and decoding methods with the use of circulant matrices of Narayana numbers and Gaussian Narayana numbers have been introduced.

 In the present paper, we obtain identities for Narayana numbers, like the sum of terms with even and odd subscripts, the sum of products of consecutive terms and the sum of squares of terms. Then, we find images DN and D2N of n-tuple NN_1,  N_2, N_3,  ...  , N_n of Narayana numbers under a map D:mathbbC rightarrow mathbbC defined as Dz_1,  z_2,  ...  ,  z_nlvert z_2-z_1rvert,  lvert z_3-z_2rvert,  ...  , lvert z_n-z_n-1rvert,  lvert z_n-z_1rvert . We are then determined the circulant, skew-circulant, and semi-circulant matrices of these images. We have been discovered Frobenius norms of these circulant matrices and relations among these norms. In addition, we find DG and D2G by taking the n-tuple GGN_1, GN_2,  ...  ,  GN_n of Gaussian Narayana numbers. After that, we create circulant, semi-circulant, and skew-circulant matrices of G,  DG,  D2G, determine their Frobenius norms, and derive relationships between them. Then, we obtain relations between norms of matrices of Narayana numbers and Gaussian Narayana numbers. Finally, coding and decoding methods with the use of circulant matrices of Narayana numbers and Gaussian Narayana numbers have been introduced.

 In the present paper, we obtain identities for Narayana numbers, like the sum of terms with even and odd subscripts, the sum of products of consecutive terms and the sum of squares of terms. Then, we find images DN and D2N of n-tuple NN_1,  N_2, N_3,  ...  , N_n of Narayana numbers under a map D:mathbbC rightarrow mathbbC defined as Dz_1,  z_2,  ...  ,  z_nlvert z_2-z_1rvert,  lvert z_3-z_2rvert,  ...  , lvert z_n-z_n-1rvert,  lvert z_n-z_1rvert . We are then determined the circulant, skew-circulant, and semi-circulant matrices of these images. We have been discovered Frobenius norms of these circulant matrices and relations among these norms. In addition, we find DG and D2G by taking the n-tuple GGN_1, GN_2,  ...  ,  GN_n of Gaussian Narayana numbers. After that, we create circulant, semi-circulant, and skew-circulant matrices of G,  DG,  D2G, determine their Frobenius norms, and derive relationships between them. Then, we obtain relations between norms of matrices of Narayana numbers and Gaussian Narayana numbers. Finally, coding and decoding methods with the use of circulant matrices of Narayana numbers and Gaussian Narayana numbers have been introduced.

Makale detayları

On the Frobenius norms of circulant matrices with Ducci sequences and Narayana and Gaussian Narayana numbers

Yazarlar:

Roji Bala | Vinod Mishra

Yayıncı:

Ankara University

Ana Başlıklar:

Algebra and Number Theory

Dergi:

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Vol.74 Issue 3, pp.460-477

Belge Tipi:

Makale

Yayın Tarihi:

9/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001357871

pISSN:

1303-5991

Narayana sequence

Gaussian Narayana sequence

Ducci map

semi-circulant matrix

Skew-circulant matrix

Sayı

Makale detayları

On the Frobenius norms of circulant matrices with Ducci sequences and Narayana and Gaussian Narayana numbers

Yazarlar: Roji Bala | Vinod Mishra

Yayıncı: Ankara University

Ana Başlıklar: Algebra and Number Theory

Dergi: Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Vol.74 Issue 3, pp.460-477

Belge Tipi: Makale

Yayın Tarihi: 9/2025

Belge Dili: İngilizce

Document ID (Org No.): 2001357871

pISSN: 1303-5991

Narayana sequence

Gaussian Narayana sequence

Ducci map

semi-circulant matrix

Skew-circulant matrix

Sayı

Yazarlar:

Roji Bala | Vinod Mishra

Yayıncı:

Ankara University

Ana Başlıklar:

Algebra and Number Theory

Dergi:

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Vol.74 Issue 3, pp.460-477

Belge Tipi:

Makale

Yayın Tarihi:

9/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001357871

pISSN:

1303-5991