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\notag
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Binomial Coefficients
Basic identities
The absorption identities.
\(\begin{align}
\binom{r}{k} &= \frac{r}{k}\binom{r-1}{k-1},\ k\in\mathbb{Z}\backslash\{0\},\\
k\binom{r}{k} &= r\binom{r-1}{k-1},\ k\in\mathbb{Z},\\
(r-k)\binom{r}{k} &= r\binom{r-1}{k},\ k\in\mathbb{Z}.
\end{align}\)
The addition identities.
\(\begin{align}
\binom{r}{k} &= \binom{r-1}{k} + \binom{r-1}{k-1},\ k\in\mathbb{Z},\\
\sum_{k\leq n}\binom{r+k}{k} &= \binom{r+n+1}{n},\ k\in\mathbb{Z},\\
\sum_{0\leq k\leq n}\binom{k}{m} &= \binom{n+1}{m+1},\ m,n\in\mathbb{N}\cup\{0\}.
\end{align}\)
The trinomial revision.
\(\begin{align}
\binom{r}{m}\binom{m}{k} &= \binom{r}{k}\binom{r-k}{m-k},\ m,k\in\mathbb{Z}.
\end{align}\)
The Vandermonde convolution.
\(\begin{align}
\sum_k\binom{r}{k}\binom{s}{n-k} &= \binom{r+s}{n}.
\end{align}\)
Stirling’s approximation
Let’s first recall the original Stirling’s formula for factorial.
-
Approximation form.
\(\begin{align}
n! \sim \sqrt{2\pi n}(\frac{n}{e})^n \, .
\end{align}\)
-
Upper/Lower bound form.
\(\begin{align}
\sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}\leq n!\leq en^{n+\frac{1}{2}}e^{-n} \, .
\end{align}\)
-
Asymptotic form.
\begin{equation}
n! = \sqrt{2\pi n}(\frac{n}{e})^n(1+O(\frac{1}{n})) \, .
\end{equation}
Apply the above Stirling’s approximation, we have:
- Upper/Lower bound for binomial coefficients.
\(\begin{align}
(\frac{n}{k})^k\leq\binom{n}{k}\leq(\frac{en}{k})^k \, .
\end{align}\)
- Entropy bound for binomial coefficients.
\((1-o(1))2^{nH(k/n)}\leq\binom{n}{k}\leq(1+o(1))2^{nH(k/n)} \, .\)
Binomial theorem
Let $x$, $y$ be arbitrary reals and $n\in\N$, we have
\begin{equation}
(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k.
\end{equation}
Multinomial theorem
Let $x_1,\dots,x_m\in\R$ and $n\in\N$, we have
\begin{equation}
(x_1+\dots+x_m)^n = \sum_{k_1+\dots+k_m=n}\binom{n}{k_1,\dots,k_m}x_1^{k_1}\cdots x_m^{k+m}.
\end{equation}
Generalized binomial theorem
Newton generalized the binomial theorem from nonnegative integers to real exponents.
Let $x$, $y$, and $r$ be arbitrary reals such that $\card{x}>\card{y}$, we have
\begin{align}
(x+y)^r &= \sum_{k=0}^{\infty}\binom{r}{k}x^{r-k}y^k\\
&= x^r + rx^{r-1}y + \frac{r(r-1)}{2}x^{r-2}y^2 + \frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots.
\end{align}
Note that as $r$ is not a nonnegative integer, we need to redefine $\binom{r}{k}$ for arbitrary $r\in\R$ as follows.
\begin{equation}
\binom{r}{k} = \frac{r(r-1)\cdots(r-k+1)}{k!} = \frac{(r)_k}{k!},
\end{equation}
where $(\cdot)_k$ is the Pochhammer symbol.
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