The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials

Abstract

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized k-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini’s Identities for the families and their polynomials. Finally, we suggest avenues for further research.

Distance Fibonacci Polynomials—Part II

“…The interest in Fibonacci-like polynomials has contributed to the emergence of many generalizations. Most of them are obtained by changing initial terms while preserving the recurrence relation (see References [17,18]) or by slight modifying the basic recursion (see References [19][20][21]). Some are obtained in the distance sense i.e., by changing distance between terms of a sequence.…”

Distance Fibonacci Polynomials by Graph Methods

“…Generalizations of numbers or polynomials of the Fibonacci type existing in the literature are mainly concentrated on combinatorial properties using generating functions or the problem of solving the recurrence relation, called a problem of Binet's formula type, see, for example, [5][6][7]. Note that some generalizations we obtain by changing initial terms and preserving the recurrence equation (see [8,9]) or by modification of the recurrence, see, for instance, [10].…”