A Note on Incomplete Fibonacci–Lucas Relations

Abstract

We define the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequently, we prove an incomplete version of the well-known Fibonacci–Lucas relation and make some extensions to the relation involving incomplete generalized bivariate Fibonacci and Lucas p-polynomials. An argument about going from the regular to the incomplete Fibonacci–Lucas relation is discussed. We provide a relation involving the incomplete Leonardo and the incomplete Lucas–Leonardo p-numbers as an illustration.

Keywords

• Fibonacci–Lucas relation
• bivariate Fibonacci p-polynomials
• incomplete generalized bivariate Fibonacci p-polynomials