Following Knuth [14, page 51], we define ( r k ) for all read r and all integers k by

step 1 It's clear so enumerate here. So we're going to do a proof. So we have n choose k is equal to n

Following Knuth [14, page 51], we define ( r k )...

Following Knuth [14, page 51], we define $\left(\begin{array}{l}r \\ k\end{array}\right)$ for all read $r$ and all integers $k$ by $$ \left(\begin{array}{l} r \\ k \end{array}\right)=\frac{r(r-1) \cdots(r-k+1)}{k(k-1) \cdots(1)}=\prod_{1 \leq j \leq k}\left(\frac{r+1-j}{j}\right), $$ when $k$ is a nonnegative integer and $$ \left(\begin{array}{l} r \\ k \end{array}\right)=0 $$ when $k$ is negative. ${ }^{20}$ Thus, $$ \left(\begin{array}{c} -7.2 \\ 2 \end{array}\right)=\frac{(-7.2)(-8.2)}{2}=29.52, $$ and $\left(\begin{array}{l}r \\ 0\end{array}\right)=1$ for all $r$, by the convention that an empty product in the definition of $\left(\begin{array}{l}r \\ k\end{array}\right)$ is one. Prove (a) $\left(\begin{array}{l}r \\ k\end{array}\right)=\frac{r}{k}\left(\begin{array}{l}r-1 \\ k-1\end{array}\right)$ if $k$ is a nonzero integer. (b) $\left(\begin{array}{l}r \\ k\end{array}\right)=\frac{r}{r-k}\left(\begin{array}{c}r-k \\ k\end{array}\right)$, when $k$ is an integer and $k \neq r$. (c) $\left(\begin{array}{l}r \\ k\end{array}\right)=\left(\begin{array}{c}r-1 \\ k\end{array}\right)+\left(\begin{array}{c}r-1 \\ k-1\end{array}\right)$, when $k$ is any integer. (d) $\left(\begin{array}{c}-r \\ k\end{array}\right)=(-1)^k\left(\begin{array}{c}r+k-1 \\ k\end{array}\right)$, when $k$ is any integer.

Following Knuth [14, page 51], we define $\left(\begin{array}{l}r \\ k\end{array}\right)$ for all read $r$ and all integers $k$ by
$$
\left(\begin{array}{l}
r \\
k
\end{array}\right)=\frac{r(r-1) \cdots(r-k+1)}{k(k-1) \cdots(1)}=\prod_{1 \leq j \leq k}\left(\frac{r+1-j}{j}\right),
$$
when $k$ is a nonnegative integer and
$$
\left(\begin{array}{l}
r \\
k
\end{array}\right)=0
$$
when $k$ is negative. ${ }^{20}$ Thus,
$$
\left(\begin{array}{c}
-7.2 \\
2
\end{array}\right)=\frac{(-7.2)(-8.2)}{2}=29.52,
$$
and $\left(\begin{array}{l}r \\ 0\end{array}\right)=1$ for all $r$, by the convention that an empty product in the definition of $\left(\begin{array}{l}r \\ k\end{array}\right)$ is one. Prove
(a) $\left(\begin{array}{l}r \\ k\end{array}\right)=\frac{r}{k}\left(\begin{array}{l}r-1 \\ k-1\end{array}\right)$ if $k$ is a nonzero integer.
(b) $\left(\begin{array}{l}r \\ k\end{array}\right)=\frac{r}{r-k}\left(\begin{array}{c}r-k \\ k\end{array}\right)$, when $k$ is an integer and $k \neq r$.
(c) $\left(\begin{array}{l}r \\ k\end{array}\right)=\left(\begin{array}{c}r-1 \\ k\end{array}\right)+\left(\begin{array}{c}r-1 \\ k-1\end{array}\right)$, when $k$ is any integer.
(d) $\left(\begin{array}{c}-r \\ k\end{array}\right)=(-1)^k\left(\begin{array}{c}r+k-1 \\ k\end{array}\right)$, when $k$ is any integer.

Key Concepts

Algebraic Manipulation of Product Expressions

The derivation of the various identities relies on algebraic manipulations of the product expressions that define the generalized binomial coefficients. By carefully canceling terms and reorganizing products, one can derive recurrence relations and other combinatorial identities. This technique is fundamental in proving formulas like the relation (r choose k) = (r/k) (r-1 choose k-1) and similar identities.

Recurrence Relations and Combinatorial Identities

Many classical combinatorial identities, such as Pascal's identity, hold true in the more general setting of generalized binomial coefficients. These recurrence relations express a binomial coefficient in terms of two or more binomial coefficients with smaller parameters. They are crucial for recursive proofs and play an important role in combinatorial arguments and in the analysis of series expansions.

Generalized Binomial Coefficients

This concept extends the classical definition of binomial coefficients from nonnegative integers to arbitrary real numbers. The definition involves using a product formula that accommodates a real (or complex) upper parameter, thereby allowing the use of binomial coefficients in contexts such as power series expansions and analytic combinatorics. The factorials in the traditional definition are replaced by analogous products that work even when the upper parameter is not a nonnegative integer.

Empty Product Convention

In the definition of the generalized binomial coefficients, the convention that an empty product is equal to one is essential. This agreement ensures that the coefficient (r choose 0) is defined to be 1 for all values of r, maintaining consistency with the classical combinatorial interpretation and supporting the validity of various binomial identities.

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