Modified Dickson

ZM^fi-V—^yy"^ (»*l) (1.D j=o " A J ) in the indeterminate^, where the symbol [ • J denotes the greatest integer function. It can be seen that z n = b W . l ) ("even), \y-pn(yA) (Koddand^O), where pn(y, 1) are the Dickson polynomials iny with the parameter c = 1 (e.g., see (1.1) of [1]). Because of the relation (1.2), the quantities Zn(y) will be referred to as modified Dickson polynomials. Information on theoretical aspects and practical applications of (usual) Dickson polynomials can be found through the exhaustive list of references reported in [1], where an extension of them has been studied. In this article we are concerned with modified Dickson polynomials taken at nonnegative integers. In fact, it is the purpose of this article to establish basic properties of the elements of the sequences of integers {Zn(k)}% (k = 0,1,2,...). More precisely, in Section 2 closed-form expressions for Zn(k) are found which, for £ = 2,3, and 4, give rise to three supposedly new combinatorial identities. Several identities involving Zn(k) are exhibited in Section 3, while some congruence properties of these numbers are established in Section 4. To obtain the results presented in Sections 2 and 3, we make use of the main properties of the generalized Fibonacci numbers U„(x) and the generalized Lucas numbers Vn(x) (e.g., see [2], [8]) defined by Un{x) = xU^{x)+U^2{xl [U0(x) = 0, Ux(x) = 1], (1.3) Vn(x) = xVn_1(x) + Vn_2(x), [F0(x) = 2,K1(x) = x], (1.4) where x is an arbitrary (possibly complex) quantity. Recall that closed-form expressions (Binet forms) for U„(x) and Vn(x) are