Editing Umbral calculus (section)

==Umbral Taylor series==
In [[differential calculus]], the [[Taylor series]] of a function is an infinite sum of terms that are expressed in terms of the function's [[Derivative|derivatives]] at a single point. That is, a [[Real-valued function|real]] or [[complex-valued function]] ''f'' (''x'') that is [[Analytic function|analytic]] at <math>a</math> can be written as: 

<math>f(x)=\sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} (x-a)^{n}</math>

Similar relationships were also observed in the theory of [[finite differences]]. The umbral version of the Taylor series is given by a similar expression involving the ''k''-th [[forward difference]]s <math>\Delta^k [f]</math> of a [[polynomial]] function ''f'',

:<math>f(x)=\sum_{k=0}^\infty\frac{\Delta^k [f](a)}{k!}(x-a)_k</math>

where

:<math>(x-a)_k=(x-a)(x-a-1)(x-a-2)\cdots(x-a-k+1)</math>

is the [[Pochhammer symbol]] used here for the falling sequential product.  A similar relationship holds for the backward differences and rising factorial.

This series is also known as the [[Finite difference#Newton's_series|''Newton series'']] or '''Newton's forward difference expansion'''.
The analogy to Taylor's expansion is utilized in the [[calculus of finite differences]].