On a functional equation associated with the trapezoidal rule

In this paper, we find the solution f1, f2, f3, f4, f5, g1: R→ R of f1(y) − g1(x) = (y − x) [f2(x) + f3 (sx+ ty) +f4 (tx+ sy) + f5(y)] for all real numbers x and y. Here s and t are any two a priori chosen real parameters. This functional equation is a generalization of a functional equation that arises in connection with the trapezoidal rule for the numerical evaluation of definite integrals and is a generalization of a functional equation studied in [10]. 1