Catalan transform of the k-Pell, k-Pell–Lucas and modified k-Pell sequence

Abstract

In this study, we present the Catalan transforms of the k-Pell sequence, the k-Pell–Lucas sequence and the Modified k-Pell sequence and examine the properties of the sequences. Then we apply the Hankel transform to the Catalan transforms of the k-Pell sequence, the Catalan transform of the k-Pell–Lucas sequence and the Catalan transform of the Modified k-Pell sequence. Also, we obtain the generating functions of the Catalan transform of the k-Pell sequence, k-Pell–Lucas sequence and Modified k-Pell sequence. Furthermore, we acquire an interesting characteristic related to the determinant of the Hankel transform of the sequences.

Special transforms of the generalized bivariate Fibonacci and Lucas polynomials

“…• If we replace p = 1, q = k in CH n (ξ, ς), HCH n (ξ, ς) we obtain the Catalan, Hankel transforms for k-Jacobsthal numbers in [23]. • If we replace p = 2, q = k in CH n (ξ, ς), HCH n (ξ, ς), CK n (ξ, ς), HCK n (ξ, ς), we obtain the Catalan, Hankel transforms for k-Pell and k-Pell-Lucas numbers in [24]. • If we replace p = 6ξ, q = −1 in b n (ξ, ς) , we obtain the binomial transforms for balancing polynomials in [30].…”

On New Polynomial Sequences Constructed to Each Vertex in an n‐Gon

“…e work can be further extended to k− Jacobsthal and Jacobsthal-Lucas numbers and k− Fibonacci and Lucas numbers. [20][21][22][23][24][25][26][27]…”

“…As in [16][17][18][19][20][21][22][23], in engineering, science, and technology, the structure of such sequences with generalizations plays a vital signi cance art and architecture. In studying such discipline, Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas polynomials have shown their signi cance as in [24][25][26][27].…”

Quaternion-Type Catalan Transforms of the ρ-Fibonacci and ρ-Lucas Numbers

“…In [13], the author introduced the Catalan transform of a given sequence and its inverse in terms of the Riordan group, and then derive many transformed sequences. By inspired of this work, the Catalan transform has been applied to many sequences; for example [18,[22][23][24][25]. In this section, we defne the quaternion-type Catalan transform and then give the generating function of the quaternion-type Catalan transform of a given sequence a n 􏼈 􏼉.…”