On the Algebraic Properties of r-circulant Matrices Associated with Generalized k-Pell-Tribonacci Numbers

Abstract: This study examines the properties of an r-circulant matrix whose entries are defined by the generalized k-Pell-Tribonacci sequence {P_k,n}. Explicit expressions are derived for the Frobenius (Euclidean) norm and the entrywise \ell_1-norm, together with closed-form formulas for the eigenvalues and the determinant of the matrix. Furthermore, upper and lower bounds for the spectral norm are established, yielding results that generalize previously reported ones corresponding to particular sequences while also providing sharper bounds for the considered norms.

• The paper establishes closed-form expressions for norms, eigenvalues, and the determinant of r-circulant matrices defined by the generalized k-Pell-Tribonacci sequence.
• It introduces a Binet-type formula for the sequence and derives summation formulas that reduce computational complexity from O(n²) to O(n).
• The work identifies necessary and sufficient invertibility criteria and highlights the matrices' relevance in coding theory and spectral analysis.

Algebraic Properties of $r$-Circulant Matrices Associated with Generalized $k$-Pell-Tribonacci Numbers

Introduction and Motivation

This work presents a thorough study of $r$-circulant matrices whose generating entries are defined via the generalized $k$-Pell-Tribonacci sequence. Third-order linear recurrence sequences, such as the Tribonacci and $k$-Pell sequences, produce integer arrays of notable combinatorial and algebraic structures. Circulant matrices, and their $r$-generalizations, are central objects in harmonic analysis, numerical linear algebra, and coding theory due to their elegant diagonalization properties and computational tractability.

While extensive literature addresses circulant and $r$-circulant matrices with entries from Fibonacci, Lucas, Pell, and their $k$-generalizations, studies combining the $k$-Pell and Tribonacci frameworks in the third-order context have not appeared. This paper addresses this lacuna by deriving closed-form analytical results for norms, spectra, and invertibility of $r$-circulant matrices generated by the generalized $k$0-Pell-Tribonacci sequence, thereby unifying and extending earlier results for special cases.

Generalized $k$1-Pell-Tribonacci Sequence: Structure and Summation

The generalized $k$2-Pell-Tribonacci sequence $k$3 is defined via the third-order recurrence

$k$4

with initial conditions $k$5, $k$6, $k$7. The sequence’s characteristic polynomial $k$8 possesses three simple roots, denoted $k$9, all parameterized by $r$0. Applying Cardano's method, the paper establishes precise intervals for the real root and shows it always dominates the asymptotic growth of $r$1 for any $r$2, a result critical for subsequent norm analysis.

A Binet-type closed formula is derived: $r$3 This compact expression is crucial for all further spectral and norm calculations.

The paper provides closed-form summation formulas for $r$4, $r$5, $r$6, and $r$7 via recursive telescoping and linear system techniques. These explicit expressions, involving only $r$8 for $r$9, significantly facilitate norm and determinant estimates for the associated matrices.

$k$0-Circulant Matrix Norms: Frobenius, Entrywise $k$1, and Spectral Norms

The $k$2-circulant matrix $k$3 is defined through the generator sequence. For these matrices, the norm analysis greatly benefits from the symmetric structure. The paper derives explicit, parameterized formulas for the Frobenius and entrywise $k$4 norms: $k$5

$k$6

These formulae reduce computational complexity from $k$7 to $k$8, leveraging the recurrence structure and index symmetry inherent in $k$9-circulant matrices.

For the spectral norm $k$0, nontrivial upper and lower bounds are established: $k$1 where $k$2, $k$3, $k$4 are closed-form sums from the combinatorial analysis above. These bounds generalize previous results for third-order Pell and Tribonacci matrix variants, providing sharper estimates, as substantiated by numerical tests (e.g., for $k$5 and moderate to large $k$6, old upper bounds are orders of magnitude above the new ones).

Spectral Properties: Explicit Eigenvalues, Diagonalization, and Determinant

The eigenstructure of $k$7-circulant matrices is central for both theoretical understanding and practical applications (e.g., for matrix functions and invertibility). This work gives a full characterization of the eigenvalues and eigenvectors: $k$8 with explicit formulas for $k$9 in terms of $r$0, $r$1, $r$2, and the roots of $r$3.

By exploiting the recurrence and constructing generating polynomials, the authors present formulae that account for cases when $r$4 coincides with the reciprocal of a characteristic root, yielding exceptional closed forms. This treatment is exhaustive and covers all spectral scenarios.

The determinant is then derived as the product of the eigenvalues, yielding the compact expression: $r$5 where $r$6 are functions of the boundary sequence values and $r$7.

Invertibility Criteria

Via algebraic and number-theoretic analysis of the generating polynomials, the paper provides necessary and sufficient conditions for invertibility of $r$8, indexed by $r$9 and the spectrum of the generating sequence. Specifically, if $r$0 for $r$1 and analogously for $r$2, the matrix is invertible. For the canonical cases $r$3 (circulant) and $r$4 (skew-circulant), invertibility is rigorously established for all $r$5, except in precisely delimited exceptional parameter configurations.

The results are further validated through computational experiments, demonstrating the presence of rare but nontrivial parameter sets resulting in singular matrices outside the established criteria.

Implications and Future Directions

The derivations unify and extend the structural theory of $r$6-circulant matrices, enabling their systematic analysis when generated by third-order recurrences beyond classical cases. The explicit forms for eigenvalues, norms, and determinant facilitate both theoretical and numerical investigations into matrix analysis, spectral algorithms, and operator theory contexts.

Practically, the results support matrix computations in spectral methods, fast algorithms via diagonalization, and broaden the class of integer-sequence structured matrices available for error correction, cryptography, and algebraic coding.

Future research may focus on:

• Generalizing to higher-order recurrences and exploring algebraic and analytic properties.
• Investigating connections with orthogonal polynomial sequences and their induced matrix algebras.
• Leveraging the explicit spectral structure for efficient computation of matrix powers, exponentials, and inverses.
• Exploring probabilistic and random matrix variants with $r$7-Pell-Tribonacci generators for applications in statistical physics and complexity theory.

Conclusion

This paper meticulously details the algebraic, spectral, and norm properties of $r$8-circulant matrices generated from the generalized $r$9-Pell-Tribonacci sequence. By deriving closed-form expressions and sharp bounds for norms and spectra, it furnishes precise tools for future analytic and algorithmic work involving structured third-order recurrence-based matrices. The results both consolidate and significantly generalize the theory of circulant-type matrices, laying a substantial foundation for ongoing investigation into structured linear algebra and its applications.

Reference: "On the Algebraic Properties of r-circulant Matrices Associated with Generalized k-Pell-Tribonacci Numbers" (2604.03817).