Generating Functions of Binary Products of Tribonacci and Tribonacci Lucas Polynomials and Special Numbers

In this paper, we introduce a new operator defined in this paper, we give some new generating functions of binary products of Tribonacci and Tribonacci Lucas polynomials and special numbers.

Incomplete generalized Vieta–Pell and Vieta–Pell–Lucas polynomials

In this paper, we introduce the incomplete Vieta–Pell and Vieta–Pell–Lucas polynomials. We give some properties, the recurrence relations and the generating function of these polynomials with suggestions for further research.

Distance Fibonacci Polynomials—Part II

In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas polynomials.

Symmetric and generating functions of generalized (p,q)-numbers

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

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

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.