The downward Clenshaw recurrence formula evaluates a sum of products of indexed coefficients by functions which obey a recurrence relation. If f(x) = sum_(k = 0)^N c_k F_k(x) and F_(n + 1)(x) = α(n, x) F_n(x) + β(n, x) F_(n - 1)(x), where the c_ks are known, then define y_(N + 2) | = | y_(N + 1) = 0 y_k | = | α(k, x) y_(k + 1) + β(k + 1, x) y_(k + 2) + c_k for k = N, N - 1, ... and solve backwards to obtain y_2 and y_1.