Incomplete Bivariate Fibonacci and Lucas 𝑝-Polynomials

Abstract

We define the incomplete bivariate Fibonacci and Lucas 𝑝-polynomials. In the case 𝑥=1, 𝑦=1, we obtain the incomplete Fibonacci and Lucas 𝑝-numbers. If 𝑥=2, 𝑦=1, we have the incomplete Pell and Pell-Lucas 𝑝-numbers. On choosing 𝑥=1, 𝑦=2, we get the incomplete generalized Jacobsthal number and besides for 𝑝=1 the incomplete generalized Jacobsthal-Lucas numbers. In the case 𝑥=1, 𝑦=1, 𝑝=1, we have the incomplete Fibonacci and Lucas numbers. If 𝑥=1, 𝑦=1, 𝑝=1, 𝑘=⌊(𝑛−1)/(𝑝+1)⌋, we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas 𝑝-polynomials are given.