ON SOME IDENTITIES FOR k-FIBONACCI DIFFERENCE SEQUENCE

Abstract

In this paper, several new identities are given for the $k$-Fibonacci difference sequence. This is accomplished by solving a class of non-homogeneous, linear recurrence relations. We address a sequence of the $k$-Fibonacci type, called the $k$-Fibonacci difference sequence. This sequence is obtained by applying the finite difference operator to the $k$-Fibonacci sequence a finite number of times. By taking advantage of the strong dependence this sequence has on its initial terms, as well as on the $k$-Fibonacci and $k$-Lucas sequences, we were able to derive a wide range of properties, including classical identities such as those of Tagiuri-Vajda, Catalan, and Cassini identities, among others. Moreover, we obtained its extension to negative indexes with explicit expressions. Several types of generating functions were also derived, including exponential and Poisson-type generating functions. In addition, we present some results concerning the limit of the ratio involving terms of the $k$-Fibonacci difference sequence for both positive and negative indexes, along with various identities for partial sums.

Keywords

• Binet’s formula
• $k$-Fibonacci sequence
• \(k\)-Lucas sequence
• $k$-Fibonacci difference sequence
• Generating function.

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