Extensions of Sury's Relation Involving Fibonacci \(k\)-step and Lucas \(k\)-step Polynomials

Based on the method of generating functions of the sequence of Fibonacci $k$-step and Lucas $k$-step polynomials, or on a crucial identity relating Fibonacci $k$-step and Lucas $k$-step polynomials, extensions of Sury's relation and the alternating Sury's relation involving Fibonacci $k$-step and Lucas $k$-step polynomials are derived, respectively. Extensions of Sury's relation involving Fibonacci-type and Lucas-type polynomials are also obtained. Of course, these relations are generalizations of the well-known Fibonacci-Lucas relation.