Algebraic Properties of Special Number Sequences

Special number sequences—such as Fibonacci, Lucas, Horadam and Jacobsthal families—exhibit rich algebraic structures that underpin a broad spectrum of mathematical theory and application. At their core lie linear recurrence relations that dictate successive terms, while closed-form expressions (often via Binet formulae) reveal deep connections with algebraic numbers. Generating functions and matrix representations furnish uniform frameworks for deriving identities, summation formulae and divisibility properties. Extensions to hybrid numbers, quaternions and polynomials have introduced multidimensional generalisations, yielding novel combinatorial and analytic insights. Minimal addition chains capture efficient exponentiation paths in computational number theory, and the study of order of appearance in sequences informs cryptographic protocols. Collectively, these algebraic perspectives sustain advances in coding theory, combinatorics, dynamical systems and theoretical physics.

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Algebraic Properties of Special Number Sequences publication trend

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Technical terms

Recurrence relation: An equation expressing each term of a sequence as a linear combination of earlier terms.

Generating function: A formal power series whose coefficients encode the terms of a sequence and facilitate algebraic manipulation.

Binet formula: A closed-form expression for terms of a linear recurrence sequence, typically involving powers of characteristic roots.

Addition chain: A sequence of integers beginning with 1, where each term is the sum of two earlier terms, used to minimise exponentiation steps.

Hybrid number: An algebraic structure generalising complex, dual and hyperbolic numbers via multiple unit elements.

Hybrinomial: A polynomial form built upon hybrid-number bases, combining sequence terms with algebraic units.

Order of appearance: The smallest index k for which a given integer divides the k-th term of a sequence, notably the Fibonacci numbers.

References

1. An Efficient Multicore Algorithm for Minimal Length Addition Chains. Computers (2019).
2. On Leonardo Pisano Hybrinomials. Mathematics (2021).
3. Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences. Advances in Continuous and Discrete Models (2020).
4. On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence. International Journal of Mathematics and Mathematical Sciences (2011).
5. A Note on Horadam Hybrinomials. Fundamental Journal of Mathematics and Applications (2022).