Is there a combinatorial identity for the following: $$\sum_{k=0}^{i}\binom{n}{k} $$ for arbitrary integers $n, i$ with $n > i$? If so, what is this identity called?
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2$\begingroup$ Alas, no, except for special values of $i$. $\endgroup$– vadim123Commented Feb 20, 2014 at 2:51
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$\begingroup$ These are values like $n$, $n/2$, yes? $\endgroup$– AyeshaCommented Feb 20, 2014 at 3:11
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$\begingroup$ Right, and also silly things like $0,1,2,3$. $\endgroup$– vadim123Commented Feb 20, 2014 at 4:33
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1 Answer
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Using a CAS, I found this "nice" expression $$\sum_{k=0}^{i}\binom{n}{k}=2^n-\binom{n}{i+1} \, _2F_1(1,i-n+1;i+2;-1) $$ May I suggest we name it, at least for the time being, Ayesha's identity.
answered Feb 20, 2014 at 6:18
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1$\begingroup$ @Ayesha. This is my pleasure and my gift to you for today ! $\endgroup$ Commented Feb 20, 2014 at 14:40