Some Properties of the (p, q)‐Fibonacci and (p, q)‐Lucas Polynomials

Abstract

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers

“…In the above generating function, for x = 1, we find the generating function of the tribonacci numbers in (3). Now, we give other special cases as the following table related to (15). Now, we define a new family of the polynomials denoted by K j := K j (x, y, z; k, m, n, c) via the generating function…”

“…In the above representation, for x = 1, we find the generating function of the tribonacci-Lucas numbers in (4). Now, we give other special cases as Table 1 and Table 2 related to (15) and (16) respectively. Table 1.…”

(P,Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class

“…Definition 1 [6] Let P(x) and Q(x) be polynomials with real coefficients. The (P, Q)-Lucas polynomials L P,Q,n (x) are defined by the reccurence relation…”

“…Definition 2 [6] Let G {Ln(x)} (z) be the generating function of the (P, Q) -Lucas polynomial sequence L P,Q,n (x).…”

Some results on convolved (p,q)-Fibonacci polynomials

“…As usual, we may also call them the Lucas u-polynomial sequence and Lucas v-polynomial sequence, respectively. Some special cases of these two polynomial sequences, which can be found in several related references, [1][2][3][4][5][6][7] are listed in Table 1 …”