Definition:Binomial Coefficient/Real Numbers
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Definition
Let $r \in \R, k \in \Z$.
Then $\dbinom r k$ is defined as:
- $\dbinom r k = \begin {cases} \dfrac {r^{\underline k} } {k!} & : k \ge 0 \\ & \\ 0 & : k < 0 \end {cases}$
where $r^{\underline k}$ denotes the falling factorial.
That is, when $k \ge 0$:
- $\ds \dbinom r k = \dfrac {r \paren {r - 1} \cdots \paren {r - k + 1} } {k \paren {k - 1} \cdots 1} = \prod_{j \mathop = 1}^k \dfrac {r + 1 - j} j$
It can be seen that this agrees with the definition for integers when $r$ is an integer.
For most applications the integer form is sufficient.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): binomial coefficient: 1.
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(3)$