Partial Sum of the Binomial Theorem

Question

The binomial theorem states $\sum_{k=0}^nC_n^kr^k=(1+r)^n$. I am interested in the function \begin{equation} \sum_{k=0}^mC_n^kr^k, \quad m<n \end{equation} for fixed $n$ and $r$, and both $m$ and $n$ are integers. Are there any notable properties for this function? Any literature references?

In particular, do any good closed-form approximations exist for this partial sum of the binomial theorem?

http://mathoverflow.net/questions/93744/estimating-a-partial-sum-of-weighted-binomial-coefficients?rq=1 http://mathoverflow.net/questions/55585/lower-bound-for-sum-of-binomial-coefficients — Douglas Zare, Mar 31 '15 at 01:07

score 3 · Answer 1 · answered Mar 31 '15 at 00:34

3

You are asking for tail estimates for the binomial distribution. An introduction can be found on Wikipedia - this has pointers to further work.

answered Mar 31 '15 at 00:34

Igor Rivin

95,560

score 2 · Answer 2 · answered Mar 31 '15 at 03:22

2

Here is one interesting property:

MR2147385
Ostrovskii, Iossif V. On a problem of A. Eremenko. Comput. Methods Funct. Theory 4 (2004), no. 2, 275–282.

answered Mar 31 '15 at 03:22

Alexandre Eremenko

88,753

Partial Sum of the Binomial Theorem

2 Answers2

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