In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number. Thus a formal power series differs from a polynomial in that it may have infinitely many terms, and differs from a power series, whose variables can take on numerical values. One way to view a formal power series is as an infinite ordered sequence of numbers. In this case, the powers of the variable are used only to indicate the order of the coefficients, so that the coefficient of x5{\displaystyle x^{5}}
is the fifth term in the sequence. In combinatorics, formal power series provide representations of numerical sequences and of multisets, and for instance allow concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring.
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms. Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value). For example, consider the series
- A=1−<!-- − -->3X+5X2−<!-- − -->7X3+9X4−<!-- − -->11X5+⋯<!-- ⋯ -->.{\displaystyle A=1-3X+5X^{2}-7X^{3}+9X^{4}-11X^{5}+\cdots .}
- B=2X+4X3+6X5+⋯<!-- ⋯ -->,{\displaystyle B=2X+4X^{3}+6X^{5}+\cdots ,}
- A+B=1−<!-- − -->X+5X2−<!-- − -->3X3+9X4−<!-- − -->5X5+⋯<!-- ⋯ -->.{\displaystyle A+B=1-X+5X^{2}-3X^{3}+9X^{4}-5X^{5}+\cdots .}
- AB=2X−<!-- − -->6X2+14X3−<!-- − -->26X4+44X5+⋯<!-- ⋯ -->.{\displaystyle AB=2X-6X^{2}+14X^{3}-26X^{4}+44X^{5}+\cdots .}
- 44X5=(1×<!-- × -->6X5)+(5X2×<!-- × -->4X3)+(9X4×<!-- × -->2X).{\displaystyle 44X^{5}=(1\times 6X^{5})+(5X^{2}\times 4X^{3})+(9X^{4}\times 2X).}
- 11+X=1−<!-- − -->X+X2−<!-- − -->X3+X4−<!-- − -->X5+⋯<!-- ⋯ -->.{\displaystyle {\frac {1}{1+X}}=1-X+X^{2}-X^{3}+X^{4}-X^{5}+\cdots .}
- [X3](B)=4,[X2](X+3X2Y3+10Y6)=3Y3,[X2</mrow
