Editing Generating function (section)

==Definition==

{{block quote
| text = A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.
| author = [[George Pólya]]
| source = ''[[Mathematics and plausible reasoning]]'' (1954) }}

{{block quote
| text = A generating function is a clothesline on which we hang up a sequence of numbers for display.
| author = [[Herbert Wilf]]
| source = ''[http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology]'' (1994)}}

===Convergence===
Unlike an ordinary series, the ''formal'' [[power series]] is not required to [[Convergent series|converge]]: in fact, the generating function is not actually regarded as a [[Function (mathematics)|function]], and the "variable" remains an [[Indeterminate (variable)|indeterminate]].  One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. Thus generating functions are not functions in the formal sense of a mapping from a [[Domain of a function|domain]] to a [[codomain]]. 

These expressions in terms of the indeterminate&nbsp;{{mvar|x}} may involve arithmetic operations, differentiation with respect to&nbsp;{{mvar|x}} and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of&nbsp;{{mvar|x}}. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of {{mvar|x}}, and which has the formal series as its [[series expansion]]; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a [[convergent series]] when a nonzero numeric value is substituted for&nbsp;{{mvar|x}}. 

===Limitations===
Not all expressions that are meaningful as functions of&nbsp;{{mvar|x}} are meaningful as expressions designating formal series; for example, negative and fractional powers of&nbsp;{{mvar|x}} are examples of functions that do not have a corresponding formal power series.