formal power series

Formal power series generalize polynomials to infinite terms. It has the following form:

A polynomial can be viewed as a formal power series in which the higher coefficients are all zeros.

However, there is a crucial difference: by having infinitely many terms, substituting a number into can diverge. However, we won’t substitute values of in the formal power series. This means that we don’t need to think about when a formal power series will ever diverge (for example, substituting for will diverge). Instead, we manipulate formal power series purely through algebraic operations, with no notion of convergence required.

Formal power series are widely used in combinatorics in the context of generating functions.

Operations

The fact that we won’t substitute a number for means that we can’t think of a formal power series as a function of numbers; instead, we need to treat it as an object in its own right. This means that we need to define their operations (though the definition and algebraic properties of those operations are quite intuitive)

Addition and Multiplication

See also: Cauchy product

If we have two formal power series, and , then their sum and product is defined as

A special case of the multiplication rule above is that multiplying by a scalar just multiplies every coefficient: .

Another special case of the multiplication is when for some . In that case we effectively shift the coefficients to the right:

This is useful for generating functions with leading zeros. For example, is the generating function of .

Quotient

Example

The geometric series is one of the most used formal power series, and its closed form can be derived just by the definitions above: Let . Then . Thus,

Derivative

The derivative of a formal power series is defined as

This definition is convenient that we can always differentiate both the formal power series and its closed form and get the expected result.

Example

with the geometric series

We can keep going:

And more generally:

From the above example, we have the following theorem:

Theorem

for any , we have

Corollary: For any , we have

Those two formulas are equivalent to each other by substitution. You would normally change the summation so that it started from , but in this case it makes no difference because for

Normal differentiation rules can also be proven, for example:

Theorem

For any formal power series and , we have

1. sum rule
2. product rule
3. quotient rule