Binomial Coefficient with One

Theorem

$\forall r \in \R: \dbinom r 1 = r$

where $\dbinom r 1$ denotes a binomial coefficient.

The usual presentation of this result is:

$\forall n \in \N: \dbinom n 1 = n$

From the definition of binomial coefficients:

$\dbinom r k = \dfrac {r^{\underline k} } {k!}$ for $k \ge 0$

where $r^{\underline k}$ is the falling factorial.

In turn:

$\ds x^{\underline k} := \prod_{j \mathop = 0}^{k - 1} \paren {x - j}$

But when $k = 1$, we have:

$\ds \prod_{j \mathop = 0}^0 \paren {x - j} = \paren {x - 0} = x$

So:

$\forall r \in \R: \dbinom r 1 = r$

$\blacksquare$

This is completely compatible with the result for natural numbers:

$\forall n \in \N: \dbinom n 1 = n$

as from the definition:

$\dbinom n 1 = \dfrac {n!} {1! \ \paren {n - 1}!}$

the result following directly, again from the definition of the factorial where $1! = 1$.

$\blacksquare$

1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients: $1.28$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(4)$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 3$: The Binomial Formula and Binomial Coefficients: Binomial Coefficients: $3.5$: $(1)$