on analytic functions, for sufficiently small . Thus, , and formally inverting the exponential yields This formula holds in the sense that both operators give the same result when applied to a polynomial. Even for analytic functions, the series on the right is not guaranteed to converge; it may be an asymptotic series. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to mentioned at the end of the section Higher-order differences. The analogous formulas for the backward and central difference operators are The calculus of finite differences is related to the umbral calculus of combinatorics. This remarkably systematic correspondence is due to the identity of the commutators of the umbral quantities to their continuum analogs ( limits), A large number of formal differential relations of standard calculus involving functions thus map systematically to umbral finite-difference analogs involving . For instance, the umbral analog of a monomial is a generalization of the above falling factorial (Pochhammer k-symbol), so that hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function in such symbols), and so on. For example, the umbral sine is As in the continuum limit, the eigenfunction of also happens to be an exponential, and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials. This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols. Thus, for instance, the Dirac delta function maps to its umbral correspondent, the cardinal sine function, and so forth. Difference equations can often be solved with techniques very similar to those for solving differential equations. The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator. Rules for calculus of finite difference operators Analogous to rules for finding the derivative, we have: Constant rule: If is a constant, then Linearity: if and are constants, All of the above rules apply equally well to any difference operator as to , including and . Product rule: Quotient rule: or Summation rules: See references. Generalizations A generalized finite difference is usually defined as where is its coefficient vector. An infinite difference is a further generalization, where the finite sum above is replaced by an infinite