OGFs, EGFs, differentiation and Taylor shifts

OGFs, EGFs, differentiation and Taylor shifts

By adamant, history, 15 months ago,

Hi everyone!

Today I'd like to write another blog about polynomials. Consider the following problem:

You're given , you need to compute .

There is a well-known solution to this, which involves some direct manipulation with coefficients. However, I usually prefer approach that is more similar to synthetic geometry when instead of low-level coordinate work, we work on a higher level of abstraction. Of course, we can't get rid of direct coefficient manipulation completely, as we still need to do e.g. polynomial multiplications.

But we can restrict direct manipulation with coefficients to some minimal number of black-boxed operations and then strive to only use these operations in our work. With this goal in mind, we will develop an appropriate framework for it.

Thanks to clyring for inspiring me to write about it with this comment. You can check it for another nice application of calculating for a specific series over the differentiation operator:

While this article mostly works with to find , there you have to calculate

to find a polynomial such that for a given polynomial .

Key results

Let and be a linear operators in the space of formal power series such that and .

The transforms and are called the Borel transform and the Laplace transform correspondingly.

As we also work with negative coefficients here, we define for , hence for such .

In this notion,

where is the differentiation operator. Thus, is a part with non-negative coefficients of the cross-correlation of and as formal power series. More generally, for arbitrary formal power series , it holds that

that is is exactly the non-negative part of the cross-correlation of and .

Detailed explanation is below.

OGF and EGF

Let be a sequence of numbers. In analytic combinatorics, there are two ways to represent it with a generating function:

Functions and are called ordinary generating function and exponential generating function correspondingly.

Example. OGF for the sequence is , while its EGF is .

Differentiation operator

The differentiation operator is formally defined as a linear operator such that . In other words,

Thus, if looked through underlying sequence perspective, on EGF corresponds to a simple shift of .

Operator exponent

Returning to our original problem, a single power of is given as

or in a symmetric form

Knowing that , we may rewrite it as

Here is an exponent of operator , formally defined as

When is constant, and commute, thus . It also means that we can get rid of and obtain which, in turn, generalizes to for an arbitrary formal power series .

Transitions

With the definitions above in mind, let's introduce operations and such that and . The operations can be computed for any power series by multiplying or dividing its coefficients with correspondingly.

There are two essential observations we should make that hold for any formal power series .

First of all, for any formal power series .

The second important observation is

which can be written generically as

for arbitrary formal power series . Here we define for , thus is for negative .

The second property, due to linearity of is further generalized as

for the arbitrary formal power series . Combining it with the first property, we get

In particular, for , we obtain

thus is the non-negative part of the cross-correlation of and .

Check your understanding

You can implement and check your solution for it on this Library Checker problem.