Sum over j, k of -1^j+k by j+j Choose k+l by r Choose j by n Choose k by s+n-j-k Choose m-j
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Theorem
Let $l, m, n \in \Z$ be integers such that $n \ge 0$.
Then:
- $\ds \sum_{j, \ k \mathop \in \Z} \paren {-1}^{j + k} \dbinom {j + k} {k + l} \dbinom r j \dbinom n k \dbinom {s + n - j - k} {m - j} = \paren {-1}^l \dbinom {n + r} {n + l} \dbinom {s - r} {m - n - l}$
Proof
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Sources
- 1994: Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science (2nd ed.): $\S 5.1$: Basic Identities $(32)$
- 1994: Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science (2nd ed.): $\S 5$: Binomial Coefficients: Exercise $83$
- 1994: Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science (2nd ed.): $\S 5$: Binomial Coefficients: Exercise $106$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $63$