Generating Function of Some k-Fibonacci and k-Lucas Sequences
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- K-Fibonacci and K-Lucas Numbers, Binet Identity, Generating Function, Convolution
- Sergio Falcon
- The k-Fibonacci and the k-Lucas numbers are particular cases of the different generalizations of the classical Fibonacci and Lucas numbers made by different authors. In this paper, first of all, we present the k-Fibonacci and the k-Lucas numbers and we remember some of the properties that we will need throughout this article. Then, we study the relationship between the product of two k–Fibonacci or k–Lucas numbers with subscripts in linear form and the k–Lucas numbers. We thus enter the main part of the paper and find the generating function of some k–Fibonacci and k–Lucas numbers, that can be used later studies. It is interesting to note that because the definition of the k-Fibonacci and the k-Lucas numbers is based on the same recurrence relation, we find that generating functions are similar for both types of numbers. As for the denominators, there is only difference in some sign, while the numerators are different because the initial conditions of both sequences are different. By last, we give examples of application of the preceding formulas to find the generation function of some new sequences.
Full text: IJISM_816_FINAL.pdf