Arithmetic indexes k- Pell ve k-Pell Lucas numberssequences

Special integer sequences are among the extensively studied topics in number theory. The thesis consists of five chapters. The fourth chapter constitutes the original part of the thesis. Research has been conducted on arithmetic-indexed generalized Pell and Pell Lucas number sequences. The first chapter provides a historical overview of the golden ratio, Pell number sequences, Fibonacci number sequences, Pell-Lucas number sequences, k-Pell, and k-Pell Lucas number sequences. The second and third chapters cover topics that contribute to the derivation of the thesis's fundamental findings. Some of these topics include arithmetic-indexed k-Fibonacci numbers, arithmetic-indexed k-Lucas numbers, arithmetic-indexed k-Jacobsthal numbers, arithmetic-indexed k-Jacobsthal Lucas numbers, and arithmetic-indexed k-Pell and k-Pell Lucas number sequences. In the fourth chapter, utilizing these topics, the definition, Binet formula, summation formula, generating function, alternative summation formulas, and proofs of several theorems are provided for arithmetic-indexed (s,t)-Pell and (s,t)-Pell Lucas number sequences, as well as arithmetic-indexed modified (s,t)-Pell number sequences. The fifth and final chapter of this thesis presents the conclusions and recommendations.