is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of of the generalization below. Note the formal correspondence of this result to Taylor's theorem. Historically, this, as well as the Chu–Vandermonde identity, (following from it, and corresponding to the binomial theorem), are included in the observations that matured to the system of umbral calculus. Newton series expansions can be superior to Taylor series expansions when applied to discrete quantities like quantum spins (see Holstein–Primakoff transformation), bosonic operator functions or discrete counting statistics. To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the Fibonacci sequence One can find a polynomial that reproduces these values, by first computing a difference table, and then substituting the differences that correspond to (underlined) into the formula as follows, For the case of nonuniform steps in the values of , Newton computes the divided differences, the series of products, and the resulting polynomial is the scalar product, . In analysis with -adic numbers, Mahler's theorem states that the assumption that is a polynomial function can be weakened all the way to the assumption that is merely continuous. Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series does not, in general, exist. The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of suitably scaled forward differences. In a compressed and slightly more general form and equidistant nodes the formula reads Calculus of finite differences The forward difference can be considered as an operator, called the difference operator, which maps the function to . This operator amounts to where is the shift operator with step h, defined by , and is the identity operator. The finite difference of higher orders can be defined in recursive manner as . Another equivalent definition is . The difference operator is a linear operator, as such it satisfies . It also satisfies a special Leibniz rule indicated above, . Similar statements hold for the backward and central differences. Formally applying the Taylor series with respect to , yields the formula where denotes the continuum derivative operator, mapping to its derivative . The expansion is valid when both sides act