OGFs, EGFs, differentiation and Taylor shifts

Правка en10, от adamant, 2022-06-20 18:57:41

Hi everyone!

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

You're given $$$P(x) = a_0 + a_1 x + \dots + a_{n-1} x^{n-1}$$$, you need to compute $$$P(x+a) = b_0 + b_1 x + \dots + b_{n-1} x^{n-1}$$$.

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 $$$g(D) f(x)$$$ for a specific series $$$g(D)$$$ over the differentiation operator:

While this article mostly works with $$$e^{aD} f(x)$$$ to find $$$f(x+a)$$$, there you have to calculate

$$$\left(\frac{D}{1-e^{-D}}\right)p(x)$$$

to find a polynomial $$$f(x)$$$ such that $$$f(x) = p(0)+p(1)+\dots+p(x)$$$ for a given polynomial $$$p(x)$$$.

Key results

Let $$$[\cdot]$$$ and $$$\{ \cdot \}$$$ be a linear operators in the space of formal power series such that $$$[x^k] = \frac{x^k}{k!}$$$ and $$$\{x^k\} = k! x^k$$$.

The transforms $$$[\cdot]$$$ and $$$\{\cdot \}$$$ are called the Borel transform and the Laplace transform correspondingly.

As we also work with negative coefficients here, we define $$$\frac{1}{k!}=0$$$ for $$$k < 0$$$, hence $$$[x^k]=0$$$ for such $$$k$$$.

In this notion,

$$$f(x+a) = e^{aD} f(x) = [e^{ax^{-1}}\{f(x)\}],$$$

where $$$D=\frac{d}{d x}$$$ is the differentiation operator. Thus, $$$\{f(x+a)\}$$$ is a part with non-negative coefficients of the cross-correlation of $$$e^{ax}$$$ and $$$\{f(x)\}$$$ as formal power series. More generally, for arbitrary formal power series $$$g(D)$$$, it holds that

$$$g(D) f(x) = [g(x^{-1})\{f(x)\}],$$$

that is $$$\{g(D) f(x)\}$$$ is exactly the non-negative part of the cross-correlation of $$$g(x)$$$ and $$$\{f(x)\}$$$.

Detailed explanation is below.


OGF and EGF

Let $$$a_0, a_1, \dots$$$ be a sequence of numbers. In analytic combinatorics, there are two ways to represent it with a generating function:

$$$\begin{gather} F(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_k x^k + \dots\\ G(x) = a_0 + a_1 x + a_2 \frac{x^2}{2} + \dots + a_k \frac{x^k}{k!} + \dots \end{gather}$$$

Functions $$$F$$$ and $$$G$$$ are called ordinary generating function and exponential generating function correspondingly.

Example. OGF for the sequence $$$1,1,\dots$$$ is $$$1+x+x^2+\dots = \frac{1}{1-x}$$$, while its EGF is $$$1+x+\frac{x^2}{2!}+\dots = e^x$$$.

Differentiation operator

The differentiation operator $$$D = \frac{d}{dx}$$$ is formally defined as a linear operator such that $$$D x^k = k x^{k-1}$$$. In other words,

$$$\begin{gather} D F = a_1 + 2 a_2 x + \dots + k a_k x^{k-1} + \dots\\ D G = a_1 + a_2 x + a_3 \frac{x^2}{2} + \dots + a_{k} \frac{x^{k-1}}{(k-1)!} + \dots \end{gather}$$$

Thus, if looked through underlying sequence perspective, $$$D$$$ on EGF corresponds to a simple shift of $$$a_0, a_1, \dots$$$.

Operator exponent

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

$$$(x+a)^k = \sum\limits_{i=0}^k \binom{k}{i} a^i x^{k-i} = k!\sum\limits_{i=0}^k \frac{a^i}{i!}\frac{x^{k-i}}{(k-i)!},$$$

or in a symmetric form

$$$\frac{(x+a)^k}{k!} = \sum\limits_{i+j=k} \frac{a^i}{i!} \frac{x^j}{j!}.$$$

Knowing that $$$\frac{d}{dx}\frac{x^k}{k!} = \frac{x^{k-1}}{(k-1)!}$$$, we may rewrite it as

$$$\frac{(x+a)^k}{k!} = \sum\limits_{i=0}^{k} \frac{a^i}{i!} \left(D^i \frac{x^k}{k!}\right) =\left(\sum\limits_{i=0}^{\infty} \frac{a^i D^i}{i!} \right) \frac{x^k}{k!} = e^{aD} \frac{x^k}{k!}.$$$

Here $$$e^{aD}$$$ is an exponent of operator $$$D$$$, formally defined as

$$$e^{aD} = \sum\limits_{i=0}^\infty \frac{(aD)^i}{i!}.$$$

When $$$a$$$ is constant, $$$a$$$ and $$$D$$$ commute, thus $$$(aD)^i = a^i D^i$$$. It also means that we can get rid of $$$k!$$$ and obtain $$$(x+a)^k = e^{aD} x^k$$$ which, in turn, generalizes to $$$e^{aD} F(x) = F(x+a)$$$ for an arbitrary formal power series $$$F$$$.

Transitions

With the definitions above in mind, let's introduce operations $$$[\cdot]$$$ and $$$\{\cdot\}$$$ such that $$$G = [F]$$$ and $$$F = \{G\}$$$. The operations can be computed for any power series by multiplying or dividing its coefficients with $$$k!$$$ correspondingly.

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

First of all, $$$\{[F]\} = [\{F\}] = F$$$ for any formal power series $$$F$$$.

The second important observation is

$$$D^i[x^j] = \frac{x^{j-i}}{(j-i)!} = [x^{j-i}],$$$

which can be written generically as

$$$D^i [f(x)] = [x^{-i}f(x)]$$$

for arbitrary formal power series $$$f(x)$$$. Here we define $$$(j-i)! = \infty$$$ for $$$i > j$$$, thus $$$[x^k]$$$ is $$$0$$$ for negative $$$k$$$.

The second property, due to linearity of $$$[\cdot]$$$ is further generalized as

$$$g(D)[f(x)] = [g(x^{-1})f(x)]$$$

for the arbitrary formal power series $$$g(D)$$$. Combining it with the first property, we get

$$$g(D) f(x) = g(D) [\{f(x)\}] = [g(x^{-1})\{f(x)\}].$$$

In particular, for $$$g(D)=e^{aD}$$$, we obtain

$$$f(x+a) = e^{aD}f(x) = [e^{ax^{-1}}\{f(x)\}],$$$

thus $$$\{f(x+a)\}$$$ is the non-negative part of the cross-correlation of $$$e^{ax}$$$ and $$$\{f(x)\}$$$.

Check your understanding

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

Теги generating function, power series

История

 
 
 
 
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  Rev. Язык Кто Когда Δ Комментарий
en10 Английский adamant 2022-06-20 18:57:41 23
en9 Английский adamant 2022-06-20 18:56:26 102
en8 Английский adamant 2022-06-20 18:53:10 6
en7 Английский adamant 2022-02-10 14:59:59 26 +Laplace transform
en6 Английский adamant 2022-02-10 14:55:23 174 explicitly mention Borel transform
en5 Английский adamant 2022-02-06 16:28:25 2
en4 Английский adamant 2022-02-06 02:46:07 2 article
en3 Английский adamant 2022-02-05 18:53:04 2
en2 Английский adamant 2022-02-05 18:52:20 17
en1 Английский adamant 2022-02-05 18:47:04 5820 Initial revision (published)