Bivariate identities related to Chebyshev and Dickson polynomials over Fq

Let Fq be a field of q elements, where q is a power of an odd prime p. The polynomial f(y)∈Fq[y] defined byf(y):=(1+y)(q+1)/2+(1−y)(q+1)/2 has the property thatf(1−y)=ρ(2)f(y), where ρ is the quadratic character on Fq. This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.