Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r/Example

Theorem

$\dfrac {x \paren {x - 2} \paren {x - 3} } {\paren {-1} \paren {-2} \paren {-3} } + \dfrac {\paren {x + 1} \paren {x - 1} \paren {x - 2} } {\paren 1 \paren {-1} \paren {-2} } + \dfrac {\paren {x + 2} x \paren {x - 1} } {\paren 2 \paren 1 \paren {-1} } + \dfrac {\paren {x + 3} \paren {x + 1} x} {\paren 3 \paren 2 \paren 1 } = 1$

Proof

This is an example of Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r:

$\ds \sum_{k \mathop = 1}^n \paren {\dfrac {\ds \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne m} } \paren {x + k - r} } {\ds \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne k} } \paren {k - r} } } = 1$

where $n = 4$ and $m = 2$.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $34$