Hyper-Leonardo Hybrinomials

Abstract

The aim of this paper is to define hyper-Leonardo hybrinomials as a generalization of the Leonardo Pisano hybrinomials and to examine some of their properties such as the recurrence relation, summation formula and generating function. Another aim is to introduce hyper hybrid-Leonardo numbers.

Hybrinomials related to hyper-Leonardo numbers

“…where r is a positive integer [10]. The hyper-Leonardo numbers have the following recurrence relation for n ≥ 1 and r ≥ 1 [10]:…”

Hyper-Leonardo Hybrinomials

“…Alp and Koçer [8] obtained some identities for the Leonardo numbers, and presented some relations among the Fibonacci, Lucas and Leonardo numbers. There are some papers on the generalization of the Leonardo numbers in the literature [9][10][11][12]. Kürüz et al [13] preferred to call the Leonardo numbers as Leonardo Pisano numbers and defined Leonardo Pisano polynomials as 𝐿𝑒 𝑛 (𝑥) = { 1, 𝑛 = 0,1 𝑥 + 2, 𝑛 = 2 2𝑥𝐿𝑒 𝑛−1 (𝑥) − 𝐿𝑒 𝑛−3 (𝑥), 𝑛 ≥ 3.…”

“…Mersin and Bahşi [11] defined hyper-Leonardo numbers 𝐿𝑒 𝑛 (𝑟) as a generalization of the Leonardo numbers, by the formula…”

“…where 𝑟 is a positive integer. Hyper-Leonardo numbers have the following recurrence relation for 𝑛 ≥ 1 and 𝑟 ≥ 1 [11]:…”