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SUMS AND PRODUCTS


This question is a variation of Exercize 1.2.3.34 from the Book The Art of Computer Programming Vol. 1, by Prof. D. Knuth, page 36.

Prove that

k=1n { ∏r=1n [x+k-r] / ∏r=1, r≠kn [k-r] } = nx .

This is a special case of the more general formula:

k=1n { ∏r=1n [Yk-Zr] / ∏r=1, r≠kn [Yk-Yr] } = ∑k=1nYk - ∑r=1nZr .

The formula can be proven using the residue theorem.

Marco Corvi - 2003