Extensions of Sury's relation involving Fibonacci \(k\)-step and Lucas \(k\)-step polynomials (Q6546719)

A relation between Fibonacci and Lucas numbers was observed by the reviewer as a consequence of a polynomial identity in 2016 [Am. Math. Mon. 121, No. 3, 236 (2014; Zbl 1541.11022)]. It turns out that this had already been noticed by \textit{A. T. Benjamin} and \textit{J. J. Quinn} [Util. Math. 56, 137--142 (1999; Zbl 0940.05008)] using a counting of colored tilings. In the present paper, the authors obtain, more generally, identities that relate Fibonacci-type and Luca-type polynomials. Their proofs are based on generating functions.