A note on the generalized bi-periodic Lucas-balancing numbers

In this study, we introduce a new class of integers called the sequence of generalized bi-periodic Lucas-balancing numbers, which extends the well-known sequence of Lucas-balancing numbers. We present several fundamental properties, including the deduction of the corresponding generating function, as well as homogeneous and non-homogeneous recurrence relations associated with this new sequence. We also formulate generalized versions of Binet's formulas for these numbers. In addition, we investigated the validity of several classical identities within this new context, such as the Tagiuri-Vajda, d'Ocagne, Catalan, and Cassini identities, considering the two different cases of the discriminant value of the equation polynomial associated with the recurrence relation. These extensions contribute to a structural and algebraic deepening of the properties of Lucas-balancing numbers.