approximation. If is three times differentiable, The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If for odd, and for even, then if it is calculated with the central difference scheme. This is particularly troublesome if the domain of is discrete. See also Symmetric derivative Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section). Higher-order differences In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for and and applying a central difference formula for the derivative of at , we obtain the central difference approximation of the second derivative of : Second-order central Similarly we can apply other differencing formulas in a recursive manner. Second order forward Second order backward More generally, the th order forward, backward, and central differences are given by, respectively, Forward or for , Backward Central These equations use binomial coefficients after the summation sign shown as . Each row of Pascal's triangle provides the coefficient for each value of . Note that the central difference will, for odd , have multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of and . Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties. Forward differences may be evaluated using the Nörlund–Rice integral. The integral representation for these types of series is interesting, because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large . The relationship of these higher-order differences with the respective derivatives is straightforward, Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order . However, the combination approximates up to a term of order . This can be proven by expanding the above expression in Taylor series, or by using the calculus of finite differences, explained below. If necessary, the finite difference can be centered about any point by