ON A GENERALIZATION OF THE PERFECT SQUARE SEQUENCE AND ITS POLYNOMIAL

Summary. In this study, we introduce a new recurrence relation of the perfect square sequence. We establish the relationship of perfect square sequence concerned with Fibonacci and Lucas sequences. We compute some important identities such as Catalan, Cassini, and special summation formula for this sequence. We associate the perfect square sequence with Lucas sequence and we call it the perfect square Lucas sequence. In addition, the Binet formula, generating function and summation formula of this sequence is obtained, as well as some properties are satisfied. Furthermore, we present the relationship of perfect square Lucas sequence with Fibonacci and Lucas sequences. Also, we obtain the relationship of perfect square Lucas sequence with perfect square sequence and we present matrix representation of this sequence. Besides, we described polynomials of perfect square and perfect square Lucas sequences. We get Binet formulas, generating functions, and Simpson formula for these polynomials. Eventually, satisfied some intriguing relations between these two polynomials, as well as we give the matrix representation of them. © 2025, University of Montenegro Faculty of Natural Sciences and Mathematics. All rights reserved.