Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2026 05 17 New generalizations and identities of Mersenne-Lucas numbers and polynomials with structural constraints 15165 10.22049/cco.2026.30674.2571 EN Kalika Prasad Department of Mathematics, Government Engineering College Bhojpur, Bihar, India 0000-0002-3653-5854 Mritunjay Kumar Singh Government Polytechnic, Nawada, Bihar, 805122, India Munesh Kumari Department of Mathematics, Government Engineering College Bhojpur, Bihar, India Journal Article 2025 05 30 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. https://comb-opt.azaruniv.ac.ir/article_15165_7bcbde788eace80a97dfa12c70346140.pdf