**Abstract:** |
A Lucas polynomial sequence is a pair of generalized polynomial sequences which satisfies the Lucas recursive relation and involves Fibonacci polynomials, Lucas polynomials, Chebyshev polynomials, and Balancing polynomials as its special cases. We give some polynomial sequence relations that are generalizations of the Sury-type identities. Two proofs are provided. One is based on an elementary identity and the other uses the method of generating functions. Secondly, we focus on the $(a,b)$-type Lucas polynomial sequence and then derive a generalized polynomial version of Melham's sums. As an application, we discuss some divisible properties of such power sums. |