New generalizations and identities of Mersenne-Lucas numbers and polynomials with structural constraints

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, Government Engineering College Bhojpur, Bihar, India

2 Government Polytechnic, Nawada, Bihar, 805122, India

Abstract

This paper introduces and investigates two new sequences, $\{R_{n}^{(k)}\}$ and $\{R_{n}^{(k)}(x)\}$, which provide a distinct generalization of the Mersenne--Lucas numbers and polynomials, respectively, where the index $n$ is expressed in the form $n = sk + r$, with $0 \le r < k$. We derive several identities for these sequences in relation to the classical Mersenne and Mersenne--Lucas numbers and polynomials. Furthermore, we examine their algebraic properties and establish connections with existing sequences and polynomial families. In addition, we obtain closed-form expressions, Cassini-type identities, partial sums, recurrence relations, and various combinatorial identities associated with these sequences.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 17 May 2026

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