Bi-periodic (p,q)-Fibonacci and Bi-periodic (p,q)-Lucas sequences

In this study, bi-periodic (𝑝𝑝, 𝑞𝑞)-Fibonacci and bi-periodic (𝑝𝑝, 𝑞𝑞)-Lucas sequences which generalize Fibonacci type, Lucas type, bi-periodic Fibonacci type and biperiodic Lucas type sequences are defined by using recurrence relations of (𝑝𝑝, 𝑞𝑞)- Fibonacci and (𝑝𝑝, 𝑞𝑞)-Lucas sequences. Binet formulas that allow us to calculate the 𝑛𝑛th terms of these sequences, generating functions are given and the convergence properties of their consecutive terms are examined. It is shown that some fundamental identities of bi-periodic (𝑝𝑝, 𝑞𝑞)-Fibonacci and bi-periodic (𝑝𝑝, 𝑞𝑞)-Lucas sequences conform to the well-known properties of Fibonacci and Lucas sequences. Also, tiling approach to the (𝑝𝑝, 𝑞𝑞)-Fibonacci and (𝑝𝑝, 𝑞𝑞)-Lucas sequences is presented. The 𝑛𝑛th (𝑝𝑝, 𝑞𝑞)-Fibonacci number is interpreted as the number of ways to tile a 1 × 𝑛𝑛 board, and 𝑛𝑛th (𝑝𝑝, 𝑞𝑞)-Lucas number is interpreted as the number of ways to tile a circular 1 × 𝑛𝑛 board by using colored 1 × 1 squares and colored 1 × 2 dominoes, where there are 𝑝𝑝 different colors for squares and 𝑞𝑞 different colors for dominoes.