Abstract
We define the incomplete bivariate Fibonacci and Lucas polynomials. In the case , , we obtain the incomplete Fibonacci and Lucas numbers. If , , we have the incomplete Pell and Pell-Lucas numbers. On choosing , , we get the incomplete generalized Jacobsthal number and besides for the incomplete generalized Jacobsthal-Lucas numbers. In the case , , , we have the incomplete Fibonacci and Lucas numbers. If , , , , we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas polynomials are given.
Highlights
Djordjevicintroduced incomplete generalized Fibonacci and Lucas numbers using explicit formulas of generalized Fibonacci and Lucas numbers in 1
Note that for the case k n−1 /2 incomplete Fibonacci numbers are reduced to Fibonacci numbers and for the case k n/2 incomplete Lucas numbers are Discrete Dynamics in Nature and Society reduced to Lucas numbers in 2
Taking x y p 1 in 2.3, we could obtain a formula for incomplete Fibonacci numbers see 2, Proposition 1
Summary
Djordjevicintroduced incomplete generalized Fibonacci and Lucas numbers using explicit formulas of generalized Fibonacci and Lucas numbers in 1. 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 1, y 1, p 1, k n − 1 / p 1 , we obtain the Fibonacci and Lucas numbers. Generating function and properties of the incomplete bivariate Fibonacci and Lucas p-polynomials are given.
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