# coefficient of a polynomial

Show that the coefficient of $$[x^nu^m]$$ in the bivariate generating function $$\dfrac{1}{1-2x+x^2-ux^2}$$ is $${n+1\choose n-2m}.$$

I tried to do this by using the multinomial theorem (an extension of the binomial theorem), which basically states that for terms $$x_1,\cdots, x_r, n\in \mathbb{N}_{\geq 0}, (x_1+\cdots + x_r)^n = \sum_{k_1+\cdots + k_r = n} \dfrac{n!}{k_1! \cdots k_r!}x_1^{k_1}\cdots x_r^{k_r}.$$

This gives that the given bivariate generating function is equal to $$\sum_{n\geq 0}(2x-x^2+ux^2)^n = \sum_{n\geq 0} \sum_{k_1+k_2 + k_3 = n} \dfrac{n!}{k_1!k_2!k_3!} (2x)^{k_1}(-x^2)^{k_2}(ux^2)^{k_3}$$.

Thus the coefficient of $$[x^n u^m]$$ should be $$\sum_{k_1 + 2k_2 = n-2m} \dfrac{(n-k_2-m)!}{k_1!k_2!m!}2^{k_1} (-1)^{k_2} .$$ I can further simplify this by replacing $$k_2$$ with $$\dfrac{n-2m-k_1}{2},$$ but I'm not sure how to get the desired result from that. Is there some other useful property of polynomials? I also realized that $$\sum_{m\geq 0} {n+1\choose n-2m} = 2^n,$$ which can be shown using Pascal's identity, though I'm not sure if this is useful.

• Technically this is not a polynomial (that would require finitely many terms of form $x^i u^j$), this is more of a bivariate generating function. – Sil Aug 28 '20 at 20:12
• @Sil sorry for the poor use of terminology. It can also be called a formal power series. – Fred Jefferson Aug 28 '20 at 20:18

$$\newcommand{\bbx}{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}{\displaystyle{#1}} \newcommand{\expo}{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}{\mathcal{#1}} \newcommand{\mrm}{\mathrm{#1}} \newcommand{\pars}{\left(\,{#1}\,\right)} \newcommand{\partiald}[]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}{\left\vert\,{#1}\,\right\vert}$$ Show that the coefficient of $$\ds{\bracks{x^{n}u^{m}}}$$ in the bivariate generating function $$\ds{1 \over 1 - 2x + x^{2} - ux^{2}}$$ is $$\ds{\bbox[5px,#ffd]{n + 1\choose n - 2m}: {\Large ?}}$$.
\begin{align} &\bbox[5px,#ffd]{\bracks{x^{n}u^{m}}{1 \over 1 - 2x + x^{2} - ux^{2}}} = \bracks{x^{n}u^{m}}{1 \over \pars{1 - x}^{2} - ux^{2}} \\[5mm] = &\ \bracks{x^{n}u^{m}}{1 \over \pars{1 - x}^{2}} \bracks{1 - {x^{2} \over \pars{1 - x}^{2}}\,u}^{-1} = \bracks{x^{n}}{1 \over \pars{1 - x}^{2}} \bracks{x^{2} \over \pars{1 - x}^{2}}^{m} \\[5mm] = &\ \bracks{x^{n - 2m}}\pars{1 - x}^{-2m - 2} = {-2m - 2 \choose n - 2m}\pars{-1}^{n - 2m} \\[5mm] = &\ {-\bracks{-2m - 2} + \bracks{n - 2m} - 1 \choose n - 2m} = \bbx{\large{n + 1 \choose n - 2m}} \\ & \end{align}
It might be more helpful to factorize the quadratic expression first (taking it as a variable in $$x$$). This way we can extract coefficient of $$x^n$$ ($$u$$ taken as a constant) and then coefficient of $$u^m$$ (in other words $$[x^n u^m]f(x,u)=[u^m]([x^n]f(x,u))$$. So, by factorization of denominator we arrive at $$\dfrac{1}{1-2x+x^2-ux^2}=\frac{1}{1-(1+\sqrt{u})x}\cdot \frac{1}{1-(1-\sqrt{u})x}$$ which by geometric series is $$(\sum_{i \geq 0}(1+\sqrt{u})^ix^i) \cdot (\sum_{j \geq 0}(1-\sqrt{u})^j x^j ).$$ So we get a coefficient of $$x^n$$ $$\sum_{k=0}^{n}(1+\sqrt{u})^k(1-\sqrt{u})^{n-k}\tag{*}$$ and the problem reduces to finding coefficient of $$u^m$$ in $$(*)$$. We can evaluate the expression for example by writing it as $$(1-\sqrt{u})^n\sum_{k=0}^{n}\left(\frac{1+\sqrt{u}}{1-\sqrt{u}}\right)^k$$ and spot the finite geometric series with $$q=\frac{1+\sqrt{u}}{1-\sqrt{u}}$$, so we can just use well-known formula for the sum $$\frac{q^{n+1}-1}{q-1}$$. After some messy algebra we get $$\frac{1}{2\sqrt{u}}[(1+\sqrt{u})^{n+1}-(1-\sqrt{u})^{n+1}],$$ which finally by Binomial theorem gives $$\frac{1}{2\sqrt{u}}\sum_{m=0}^{n+1}\binom{n+1}{m}\sqrt{u}^{m}(1-(-1)^{m}).$$ For even $$m$$ the terms vanish and we are left with $$\sum_{m=0}^{\lfloor n/2 \rfloor}\binom{n+1}{2m+1}u^{m}.$$ Now just read off the coefficient, also perhaps use $$\binom{n}{k}=\binom{n}{n-k}$$.