Timeline for Binomial ID $\sum_{k=m}^p(-1)^{k+m}\binom{k}{m}\binom{n+p+1}{n+k+1}=\binom{n+p-m}{n}$

5 events

when toggle format	what		by	license	comment
Dec 5, 2016 at 14:25	comment	added	Noam D. Elkies		I suppose you can also use a (formal) differential equation to prove this.
Dec 5, 2016 at 12:38	vote	accept	Matt Majic
Dec 5, 2016 at 5:54	comment	added	Fedor Petrov		@Noam but why formal series $(1+t)^x=\sum \binom{x}{n} t^n$ satisfy this equality? It may be justified by some abstract nonsense like 'this is known for reals or for positive integers, but the coefficients are polynomials in $x,y$, thus it holds formally too.' This argument may be reduced to bit less abstract: 'both parts of CV are polynomials in $x,y$ of degree at most $\ell$, thus it suffices to check that their values agree for integers $x,y\geqslant 0, x+y\leqslant \ell$ (this triangle is an interpolating set for polynomials of degree at most $\ell$) , where identity is just obvious.'
Dec 5, 2016 at 4:32	comment	added	Noam D. Elkies		Right, with the same generatingfunctionological explanation: $(1+t)^x (1+t)^y = (1+t)^{x+y}$.
Dec 5, 2016 at 4:15	history	answered	Fedor Petrov	CC BY-SA 3.0