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

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Titel: On the Frobenius norms of circulant matrices with Ducci sequences and Narayana and Gaussian Narayana numbers.
Autoren: BALA, Roji, MISHRA, Vinod
Quelle: Communications Series A1 Mathematics & Statistics; 2025, Vol. 74 Issue 3, p460-477, 18p
Schlagwörter: CIRCULANT matrices, MATRIX norms, MATRICES (Mathematics), COMBINATORICS, DECODING algorithms
Abstract: 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 N = (N1, N2, N3, ..., Nn) of Narayana numbers under a map D: C → C defined as D(z1, z2, ..., zn) = (|z2 - z1|, |z3 - z2|, ..., |zn - zn-1|, |zn - z1|). 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 G = (GN1, GN2, ..., GNn) 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. [ABSTRACT FROM AUTHOR]
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Abstract: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 N = (N<subscript>1</subscript>, N<subscript>2</subscript>, N<subscript>3</subscript>, ..., N<subscript>n</subscript>) of Narayana numbers under a map D: C → C defined as D(z<subscript>1</subscript>, z<subscript>2</subscript>, ..., z<subscript>n</subscript>) = (|z<subscript>2</subscript> - z<subscript>1</subscript>|, |z<subscript>3</subscript> - z<subscript>2</subscript>|, ..., |z<subscript>n</subscript> - z<subscript>n</subscript>-<subscript>1</subscript>|, |z<subscript>n</subscript> - z<subscript>1</subscript>|). 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 G = (GN<subscript>1</subscript>, GN<subscript>2</subscript>, ..., GN<subscript>n</subscript>) 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. [ABSTRACT FROM AUTHOR]
ISSN:13035991
DOI:10.31801/cfsuasmas.1514790