Polynomial Expansions (useful formulas)

Let $\mathrm{A}$ be the set of all " $\color{#3D99F6}{\text{useful}}$ " things.

Let $\mathrm{B}$ be the set of all " $\color{#20A900}{\text{Awesome}}$ " things.

Let $\mathrm{C}$ be the set of all " $\color{#D61F06}{\text{fascinating}}$ " things.

Let $\mathrm{D}$ be the set of all " $\color{#624F41}{\text{easily understandable}}$ " things.

What this note contains is an element of $\mathrm{A \cap B \cap C \cap D}$ ....

$\color{#3D99F6}{\textbf{Polynomial Expansions}}$

$\mathbf{1.}\quad \displaystyle \dfrac{1-x^{m+1}}{1-x} = 1+x+x^2+...+x^m = \sum_{k=0}^m x^k$

$\mathbf{2.} \quad \displaystyle \dfrac{1}{1-x} = 1+x+x^2+... = \sum_{k=0}^\infty x^k$

$\mathbf{3.}\quad \displaystyle (1+x)^n = 1+\binom{n}{1} x + \binom{n}{2}x^2+...+\binom{n}{n}x^n = \sum_{k=0}^n \dbinom{n}{k} x^k$

$\mathbf{4.}\quad \displaystyle (1-x^m)^n = 1-\binom{n}{1}x^m+\binom{n}{2}x^{2m}-...+(-1)^n\binom{n}{n} x^{nm}$

$\quad \quad \quad \quad\quad \displaystyle = \sum_{k=0}^n (-1)^n \dbinom{n}{k}x^{km}$

$\mathbf{5.}\quad \displaystyle\dfrac{1}{(1-x)^n} = 1+ \binom{1+n-1}{1} x + \binom{2+n-1}{2} x^2+...+\binom{r+n-1}{r} x^r+......$

$\quad \quad \quad \quad \quad \displaystyle =\sum_{k=0}^\infty \dbinom{k+n-1}{k} x^k$

$\color{#69047E}{\text{Tremendously useful}}$ in calculating the $\color{#3D99F6}{\text{co-efficient}}$ of any term in specially generating functions that we come across, in many combinatorics problems...

Taken From - Alan Tucker's "Applied Combinatorics"

Good luck problem solving !

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Polynomial Expansions (useful formulas)

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