Generating series subtraction

Question

If after subtracting two generating series, we get an expression such as:

$1−(1/x^2)$

does this mean that the resulting series has coefficient 0 for all (x, x^2, x^3, x^4,....) or is it not even considered a valid one because the x^2 is in the denominator?

Answer 1

We consider an (ordinary) generating series $A(x)$ to be an expression of the form \begin{align*} A(x)&=\sum_{j=0}^\infty a_jx^j\\ &=a_0+a_1x+a_2x^2+a_3x^3+\cdots \end{align*}

The difference $A(x)-B(x)$ of two generating series has the form \begin{align*} A(x)-B(x)&=\left(a_0+a_1x+a_2x^2+\cdots\right)-\left(b_0+b_1x+b_2x^2+\cdots\right)\\ &=\left(a_0-b_0\right)+(a_1-b_1)x+(a_2-b_2)x^2+\cdots \end{align*} We observe the difference of two (ordinary) generating series is again an ordinary generating series with terms $a_jx^j$, $j$ non-negative integers.

The expression \begin{align*} 1-\frac{1}{x^2}=1-x^{\color{blue}{-2}}\tag{1} \end{align*} has $\color{blue}{-2}$ as exponent which is negative. So, this is no generating series at all and can't be the result of a subtraction of two generating series. It seems there was a wrong calculation leading to (1).

Answer 2

$$1-(1/x^2)$$ is not a valid power series because it does not have a well defined constant coefficient (i.e. the coefficient of $x^0$).