ON THE 𝒌-VIETA-PELL AND 𝒌-VIETA-PELL-LUCAS SEQUENCES

The aim of this paper is to define the 𝑘-Vieta-Pell and 𝑘-Vieta-Pell-Lucas sequences, and some terms of these sequences are given. Then, we find the relations between the terms of the 𝑘-Vieta-Pell and 𝑘- Vieta-Pell-Lucas sequences. Also, we give the summation formulas, generating functions, etc. We also derive the Binet formulas using two different approaches. The first is in the known classical way and the second is with the help of the sequence's generating functions. Moreover, we calculate the special identities of these sequences like Catalan and Melham. Finally, we examine the relations between the 𝑘-Vieta-Pell sequence and various other sequences, including Fibonacci, Pell, and Chebyshev polynomials of the first kind. Similarly, we analyze the 𝑘-Vieta-Pell-Lucas sequence in relation to Lucas, Pell-Lucas numbers, Chebyshev polynomials of the second kind, and other sequences. In addition, for special 𝑘 values, these sequences are associated with the sequences in OEIS.